$B    E7fi    fib? 


Vf 


IN  MEMORIAM 
FLORIAN  CAJORl 


V 

I 


I 


^-'^^      vs. 


^J 


F 


C/^t^a<.^^   CLgi^l.^-^ 


ELEMENTS 


OF 


ARITHMETIC, 

THEORETICAL  AND  PRACTICAL; 


ADAPTED  TO  THE  USE  OF  SCHOOLS, 
AND  TO  PEIVATE  STUDY. 


BY  F.  n.  MA88LER.    F.  A.  P.  S. 


xrEW-YORs: 

l*ftlNTED  AND  PUBLISHED  BY  JAMES  BLOOMFIELB. 

1826. 


SOUTHERLY  DISTRICT  OF  KEW-YORK,  SS. 

BE  IT  REMEMBERED,  That  on  the  6th  day  of  Oc- 
L.S.     tober,  A.  D.  1826,  in  the  51st  year  of  the  ludependence 

of  the  United  Slates  of  America,  F.  R.  HASSLER,  of 
the  said  District,  hath  deposited  in  this  oflBce  the  title  of  a  Book, 
the  right  whereof  he  claims  as  Author,  in  the  words  following;, 
to  wit : 

Elements  of  AriihmttiCf  Theoretical  and  Practical ;  adapted  to 
tht  use  of  Schools,  and  to  Private  Studi/.  By  F.  R.  HASSLER, 
F.  A.  P.  S, 

la  conformity  to  the  Act  of  Congress  of  the  United  States,  en- 
titled "  An  Act  for  the  encouragement  of  Learning,  by  securing 
the  copies  of  Maps,  Charts,  and  Books,  to  the  authors  and  pro- 
prietors of  such  copies,  during  the  time  therein  mentioned."  And 
also  to  an  Act,  entitled  "  An  Act,  supplementary  to  an  Act,  enti- 
tled an  Act  for  the  encouragement  of  Learning,  by  securing  the 
copies  of  Maps,  Charts,  and  Books,  to  the  authors  and  proprietors 
of  such  copies,  during  the  times  therein  mentioned,  and  extending 
the  benefits  thereof  to  thfi  arts  of  designing,  engraving,  and  etch- 
ing historical  and  o(,her  prints.*' 

JAMES  DILL, 
Clerk  of  the  Southern  District  of  M^v-York. 


CAJCRI 


INTRODUCTION 


Arithmetic  contains  the  first  elements  of  rea- 
soning upon  quantity  ;  its  principles  take  their  rise 
in  ideas  so  simple  as  to  he  adapted  to  the  most  un- 
tutored mind,  and  to  the  lowest  capacity.  It  is  at 
the  same  time  so  indispensahle  for  every  human 
heing,  not  only  in  common  life,  but  in  the  pursuits 
of  the  highest  sciences,  that  it  forms  the  most  pro- 
per, and  has  always  formed  one  of  the  principal 
branches  of  the  earlier  education  of  youth. 

By  its  very  nature  it  furnishes  the  means  of  de- 
veloping the  reasoning  faculties,  from  the  time  of 
their  first  beginning  to  expand  themselves,  and  of 
habituating  them  to  correctness  and  precision.  It 
therefore  gives  the  human  mind  the  power  and  dis- 
position to  reason  upon  sound  and  correct  principles. 

It  is  therefore  the  duty  of  the  faithful  teacher  of 
youth,  (not  the  mere  teacher  for  his  own  private 
emolument,)  to  take  advantage  of  this  property  of 
arithmetic,  and  apply  it  to  cultivate  the  mind,  and 
enligiiten  the  understanding  of  his  scholars,  by  a 
proper  reasoning  in  this  elementary  science ;  he 
should  not  make  it  the  object  of  the  memory  alone  ^ 
a  method  that  leaves  no  impression  upon  the  mind^ 
whose  results  are  therefore  lost  again  as  soon  as 
the  school  is  dismissed. 

To  neglect  to  take  this  advantage  of  the  study  of 
arithmetic,  is  either  a  proof  of  ignorance,  or  an  ac- 
tual dereliction  of  duty.  This  may  appear  strong 
to  many  people,  but  strength  is  the  essential  pro- 
perty of  truth*     I  can  safely  appeal  to  those  who 

9181' *-i 


IV  INTRODUCTIOBT. 

have  in  early  youth  been  taught  by  the  negligent 
method  of  mere  rules,  and  have  at  a  later  period 
attained  scientific  eminence,  to  decide  between  this 
and  any  contrary  assertion. 

The  difficulties  that  the  young  experienc6  on  en- 
tering upon  any  scientific  studies,  in  colleges,  or 
otherwise,  are  well  known ;  the  path  to  be  followed 
there,  must  be  that  of  reasoning,  and  no  prepara- 
tions are  made  for  this  by  their  previous  education, 
for  the  cultivation  of  the  memory  alone,  is,  from  the 
very  constitution  of  the  human  mind,  always  detri- 
mental to  the  reasoning  faculty. 

However  the  opportunity,  as  has  been  stated, 
exists,  of  cultivating  the  reasoning  faculty  at  an 
earlier  period,  by  familiarizing  the  scholar  with 
the  simple  reasonings  of  elementary  arithmetic. 
The  step  from  that  to  higher  or  general  arithmetic, 
usually  called  Mgebra,  becomes  by  this  mode,  both 
short  and  simple,  as  in  its  nature  it  really  is ;  and 
the  scholar  who  does  not  wish  to  go  farther  than 
common  arithmetic,  can  alone  obtain  the  knowledge 
of  the  propriety  or  principles  of  its  application  to 
any  occurrence  in  common  life,  by  a  knowledge  of 
it,  founded  upon  correct  reasoning.  It  is  entirely 
wrong  to  say  and  act.  upon  the  ground,  *'  Iwant 
to  know  how  to  do  this  or  that,"  the  principle  must 
be,  *<  I  wish  to  understand  this  or  that,"  if  ever  any 
lasting  good  result  shall  be  obtained. 

My  object  in  undertaking  this  work  was  not  to 
swell  the  number  of  elementary  treatises  on  arith- 
metic, but  may  be  stated  as  follows. 

1st.  I  wish  to  smooth  the  path  of  the  teacher  and 
the  scholar,  by  explaining  and  proving,  the  pro- 
priety and  correctness  of  any  step  that  is  taken,  by 
previous  reasonings,  leading  to  the  discovery  of  the 
principle  that  ought  to  direct  it,  and  therefore  point- 
ing out  the  rule  for  the  appropriate  operation  ;  and  I 
have,  therefore,  not  been  content  to  give  the  final 


INTRODUCTIOlf.  V 

result  alone,  and  the  example  for  its  proof,  which 
is  an  individual,  and  consequently  a  defective  me- 
thod, while  reasoning  always  leads  to  general  pro- 
positions and  proofs.  In  this  way  we  attain,  step 
by  step,  to  the  real  scientific  structure  of  this  ele- 
mentary science,  and  thus  all  the  operations  become 
satisfactory  to  the  mind,  and  therefore  agreeable  to 
the  growing  intellect  of  the  scholar. 

In  carrying  such  a  system  through  the  whole 
extent,  to  that  point  where  more  general  and  exten- 
sive considerations,  of  a  higher  analytic  nature,  arc 
to  guide  us,  1  have  even  thought  it  possible  to  make 
a  treatise  which  a  man  of  science  might  look  at 
with  some  satisfaction,  and  by  which  the  young 
scholar  would  arrive  at  the  entrance  of  his  higher 
scientific  studies,  properly  prepared  by  a  correct 
habit  of  reasoning. 

2d.  The  young  and  untutored  mind,  in  truth, 
reasons  analytically  ;  a  boy,  and  in  fact  a  man,  asks 
always  WHY;  and  as  he  enters  more  and  more  deep- 
ly into  the  investigation,  continues  to  ask  the  rea- 
son of  every  thing  that  is  said  to  him  in  the  way  of 
explanation.  The  reason  of  this  lies  in  the  nature 
of  his  situation  ;  he  cannot  proceed  synthetically, 
because  synJhesis  needs  some  previous  data,  averr- 
ed, given,  or  adopted,  on  which  to  build  the  reason- 
ing to  arrive  at  a  conclusion.  This  does  not  y&t 
exist  at  this  early  stage  of  instruction. 

In  following  this  mode,  and  grounding  every 
conclusion  upon  inquiry,  of  whicli  the  ground  lies, 
cither  in  the  human  mind  itself,  even  untutored,  or 
in  the  result  of  preceding  investigations,  I  intend 
to  make  a  book  which  a  lad  remote  from  cities, 
although  he  might  not  have  had  the  benefit  of  a 
good  early  education,  can  take  in  hand  usefully, 
and  which  a  simple  knowledge  of  reading,  coupled 
with  his  own  desire  for  improvement  and  instruc- 
tion, would  induce  him  to  take  up,  and  undertake 
1  # 


VI  INTRODUCTION. 

to  study,  as  both  useful  and  agreeable ;  useful,  be- 
cause it  would  show  him  the  means  of  accounting 
to  himself  for  the  result  of  his  own  labours ;  and 
agreeable,  because  it  would   afford  him  a  pleasing 
object  of  speculation  for  his  winter  evenings.     I 
should  be  delighted  to  see  several  such  lads,  pass- 
ing an  evening  together,  with  this  book  between 
them,  each  his  slate  and  pencil  before  him,  discuss- 
ing,  mutually  giving  and  solving,  the  questions 
which  they  learn  from  it  to  form  out  of  the  occur- 
rences around  them.     I  can  promise  them  more  sa- 
tisfaction from  it,  than  in  their  passing  that  time  in 
the  bar-room  of  a  public  house,  or  a  grocery  ;  and 
more  beneficial,  economical  results,  from  the  ex- 
penditure in  book,  slate,  and  pencil,  to  assist  their 
studies,    (for  they  must  write  every  thing,)  than 
were  they  to  lay  out  the  cost  in  the  vile  liquor,  that 
emptiness  of  mind  leads  them  to  call  for ;  they  will 
soon  be  able  to  calculate :  that  they  even  make  a 
saving,  if  they  write  their  full  studies,  ideas,  and 
questions,  on  paper,  with  pen  and  ink,  in  comparison 
with  the  expences  of  the  deleterious  pleasures  of  a 
bar-room.     If  I  should  succeed  only  in  this  part  of 
my  aim,  I  would  consider  my  labour  as  sufficiently 
rewarded;   and  I  would  have  the  greatest  enjoy- 
ment, to  meet  witli  such  a  company,  afford  them 
assistance,  and  partake  of  their  rational  amusement. 
For  the  use  of  this  book,  I  should  like  to  advise, 
the  teacher,  as  well  as  the  student,  first  to  peruse 
attentively  the  theoretical  principles  of  any  rule  or 
subject,  and  then  exercise  his  scholars,  or  himself, 
in  the  application,  which  will  give  him  an  opportu- 
nity to  generalise,  and  clear  up  their,  or  his,  ideas 
properly  ;  and  after  having  gone  through  any  of  the 
principal  subdivisions,  to  take  a  general  view  of  the 
whole ;  taking  care  to  comprehend  the  leading  prin- 
ciples, and  the  mode  of  considering  the  subject,  that 
has  been  treated  of;  in  this  way  he  will  be  enabled 


UTTRODrCTIOK.  Vll 

to  make  a  proper  use  of  it  in  the  parts  to  be  treated 
next. 

It  is  an  unavoidable  condition  in  every  systematic 
work,  that  the  subsequent  parts  shall  be  grounded 
upon  the  preceding  ones,  and  therefore  these  must  be 
supposed  known  in  the  progress  of  the  work,  as  it 
proceeds.  Therefore  also  the  study  of  no  systematic 
and  good  work,  can  be  begun  in  any  other  part  than 
at  the  beginning,  by  any  scholar ;  that  is,  a  per- 
son not  fully  acquainted  with  the  wliole  subject  of 
the  book,  but  seeking  instruction  from  it.  If  any 
person  thinks  he  knows  already  some  of  the  ele- 
mentary parts,  and  wishes  to  study  only  tbe  sub- 
sequent part,  it  is  necessary  for  him  to  read  over, 
attentively,  the  parts  with  which  he  is  acquainted ; 
to  make  himself  acquainted  with  the  manner  in  which 
the  author  expresses  himself  upon  those  subjects, 
which  he  has  his  own  ideas  upon.  By  comparing 
these  together,  he  will  be  able  to  understand  pro- 
perly, afterwards,  those  parts  with  which  he  is  not 
acquainted^  and  therefore  read  and  study  with 
success ;  which  otherwise  will  certainly  not  be  the 
case.  This  is  nothing  else  but  what  is  necessary 
between  all  men,  in  any  intercourse,  that  is,  the 
necessity  of  being  acquainted  with  each  others' 
language. 

J\'<:tc-York,  October,  1826. 

F.  R.  HASSLER. 


PAR T  r 


FIEST  ELEMENTS    AND   DEDUCTION   OV  THE    FOUR 
BULES  or  ARITHMETIC. 

CHAPTER  J. 

Fundamental  Ideas  of  Q^iiantity, — System  of 
JVumeration, 

§  1.  QUANTITY,  which  is  the  object  of  Aritli* 
metic,  is  the  idea  that  has  reference  to  any  thing 
whatever,  arising  from  the  consideration  of  its  be- 
ing susceptible  of  being  more  or  less ;  without  re- 
gard to  the  nature  or  kind  of  the  thing  itself.  It  is 
not  therefore  an  absolute  existence ;  but  a  relative 
idea,  that  can  be  referred  to  any  object  whatever. 

§  2.  No  quantity  therefore  can  be  called  great 
or  small,  much  or  little,  in  itself;  it  can  be  so  only 
in  relation  to  another  quantity  of  the  same  kind, 
which  would  be  smaller  or  greater. 

§  3.  Objects  of  diiferent  kinds  cannot  be  com- 
pared with  each  other  directly  by  their  quantity  only. 
When  therefore  Objects  of  diiferent  kinds  are  to  be 
considered  in  Arithmetic,  it  becomes  necessary  : 
that  a  certain  relation  be  given  between  them,  which 
is  completely  arbitrary  as  to  quantity  itself,  and 
must  be  determined  before  any  comparison  can  take 
place. 

§  4.  The  mutual  relation  of  quantities  to  each 
other,  under  certain  given  conditions,  is  the  object  of 
arithmetic.  In  this  general  acception  then  it  ad- 
mits any  number  of  systems  of  combination,  that 
the  imagination  can  devise. 


10  FUNDAMENTAL   IDEAS    OF   IJUANTITT. 

§  5.  To  forn<  a  cSesir  and  distinct  idea  of  arith- 
metic it  is  necessary,  to  impress  the  mind  fully  with 
these  fiindamer/tal  ideas,  and  the  general  principles 
that  r&liii>v  from  thsm.  By  comparing  every  ope- 
ration of  arithmetic  with  them,  they  will  become 
always  more  and  more  clear  and  useful ;  the  whole 
system  of  arithmetic  will  become  the  more  simple,  the 
more  its  principles  are  generalized. 

$  6.  Common  ai'ithmetic,  which  might  also  be 
called  with  propriety,  determinate  arithmetic,  limits 
itself  to  the  most  simple  combinations  of  quantities, 
and  these  are  all  grounded  successively  upon  the 
first  elementary  idea  of  increase  or  decrease,  or 
more  or  less,  either  simple  or  repeated  successively, 
or  according  to  certain  determined  laws. 

§  7.  To  express  quantities  we  make  use,  in  our 
system  of  common  arithmetic,  of  ten  figures  only, 
by  the  means  of  which,  and  by  their  relative  places, 
according  to  a  certain  law,  we  can  express  any 
quantity  whatsoever.  This  law  is  called  the  sys- 
tem of  numeration  5  and  in  particular  the  decimal 
system,  from  the  individual  circumstance,  of  its 
using  ten  different  figures,  nine  of  which  are  signifi- 
cant, and  the  tenth  indicates  the  absence  of  the 
quantity  (or  thing,  or  object.) 

5  8.  These  figures  are  in  regular  succession 
1,  2,  3,  4,  5,  6,  7,  8,  9,  and  0;  this  last  is  used  to  de- 
note the  absence  of  a  quantity ;  the  1,  denotes  the 
unit  of  any  object,  of  whatever  kind  or  nature  it 
may  be ;  the  subsequent  denote  in  regular  succes- 
sion each  one  object  more  than  the  one  before  it. 

§  9.  To  denote  quantities  which  exceed  the  num- 
ber of  significant  figures,  (or  above  9,)  recourse  is 
had  to  a  law  that  assigns  superior  values  to  these 
figures,  according  to  the  order  in  which  they  arc 
placed,  assigning  to  them  a  value  as  many  times 
greater,  in  every  successive  change  of  pi  ace  from  the 
right  to  the  left,  as  the  number  of  figures  indicates, 


NUMEKATIOX .  H 

and  therefore  in  our  usual  system  of  ten  figures,  a 
tenfold  value.  This  must  itecesvsaiily  be  the  law  if 
the  system  be  able  to  express  all  numbers,  because 
any  other  law  giving  another  relation  of  value  to 
the  places,  tliaii  the  number  of  figures,  would  either 
leave  a  space  of  quarstity  unexpressed,  or  occasion 
double  expressions,  if  it  were  to  increase  in  a 
greater  or  less  ratio  than  that  number.  (The  cir- 
cumstance of  this  increase  taking  place  from  the 
right  towards  the  left  originates  in  the  fact,  that  this 
system  is  borr'owed  from  the  Arabic  or  rather 
Asiatic  nations,  who  have  the  habit  of  writing  from 
the  right  towards  the  left,  instead  of  our  writing 
from  the  left  towards  the  right.) 

§  1 0.  Thence  we  have  for  the  successive  values  of 
the  numbers,  in  their  successive  places  from  th« 
right  to  the  left,  the  denominations  shown  in  the  fol- 
lowing table : 


C    (t> 

a   3 


Ci-  O    2    3  EJ 


-i 


^3         H  ,-*    x    "1    '"-' 

et       o2a.s.D        Q.D 

f3'-i=ffi'i2iiil?P" 

•      •      CO    •  C  Ci.  D-  CO 

03    Cft    ai    • 

1      1     1,   1     1     1,1     1     1,   1     1     1,1     1 

Such  would  be  the  value  or  denomination  of  any 
figure,  placed  irs  any  one  of  the  places,  and  if  no 
quantity  of  one  or  the  other  of  these  denominations 
is  to  be  expressed,  the  place  of  it  must  be  supplied 
with  a  0,  in  order  to  give  to  the  next  figure  its  pro- 
per rank._ 


12  NtlMERATIOIf. 

§  11.  In  reading  the  numbers  we  follow  our  usual 
way  of  reading,  and  therefore  express  the  great- 
est quantities  first;  to  render  this  reading  more 
easy  it  is  also  customary  in  large  numbers  :  to  di^ 
vide  off  by  a  (,)  every  three  figures,  which  divides 
them  by  hundreds  ;  so  for  example  : 

689,347  would  read  thus: 
Six  hundred  and  eighty-nine  thousands^  three  hun- 
dred and  forty-seven,  (understood  units.) 

13,842,167  reads  thus: 
Thirteen   millions^    eight   hundred    and  forty-tw© 
thousands^  one  hundred  and  sixty-seven.     (It  will 
be  proper  in  this  way  to  exercise  the  beginner  in 
reading  numbers,  or  what  is  called  numeration.) 

§  12.  It  may  be  easily  conceived  :  that  other  sys- 
tems might  be  formed  upon  the  same  principles,  and 
of  course  with  the  same  properties,  as  to  the  ex- 
pression of  the  greater  quantities  by  the  successive 
rank  or  place  of  the  figures,  and  with  any  other 
greater  or  smaller  numbeV  of  significant  figures  | 
besides  the  (0,)  which  must,  like  the  unit,  make  part 
of  every  such  system  of  numeration. 

If  no  other  figures  were  used  but  (I)  and  (0,)  that 
is  presence  or  absence  of  the  quantity  indicated  by 
any  rank  or  place  of  figure,  the  value  in  each  place 
will  always  be  successively  double  of  that  in  the  pre- 
ceding place,  and  the  whole  of  the  calculation  would 
become  a  mechanical  mutation  of  places ;  so  for  in- 
stance in  this  system  the  following  numbers  111101, 
transcribed  into  our  usual  decimal  system,  would  be 
32,  16,  8,  4,  and  1:  or  (61.)  (It  will  be  a  very 
good  exercise  for  the  reflection  of  the  scholar  to  try 
some  of  this  kind  of  expressions  in  different  systems.) 
§  IS.  It  may  also  assist  in  clearing  up  the  prin- 
ciples of  the  decimal  system >  to  contrast  it  with  the 
old  Roman  system  of  numeration.  This  consists 
in  the  use  of  seven  letters  having  each  a  particular 
signification^  as : 


I 


BTUMEKATION,  &C.  1$ 

M,  for  one  thousand 
D,  —  five  hundred 
C,  —  one  hundred 
L,  —  fifty 
X,  —  ten 
V,  —  five 
I,    —  unity. 

In  this  system  therefore,  the  numeration  consistt* 
merely  in  writing  as  many  of  these  letters  as  will 
make  out  the  quantity  desired  ;  and  the  whole  arith- 
metic consisted,  in  placing  or  taking  away,  upon  a 
black  board,  as  many  marks  under  the  denomination 
of  each  of  these  letters,  as  the  calculation  required.  A 
bad  habit  of  the  Romans,  m  later  ages,  introduced  in- 
to this  system  anomalies  arising  from  their  consid- 
ering one  of  the  figures  of  an  inferior  number,  when 
placed  before  a  higher  one,  as  subtracted  or  taken 
away,  from  it,  as  for  example ;  IV  was  written  for 
four,  XC  for  ninety,  and  so  on. 


CHAPTER  II. 

General  Ideas,  and  JSTotation   of  the  Four  Rules  oj 
•Arithmetic, 

§  14.  The  first  and  simplest  combination  of  quan- 
tities, and  therefore  also  the  first  and  simplest  ope- 
ration in  arithemetic,  from  which  all  others  pro- 
ceed, is  called  addition.  Of  this  we  have  already  an 
example  of  the  simplest  kind  in  the  system  of  fig- 
ures, that  presents  the  successive  additions  of  unity 
in  their  regular  order  of  succession,  and  therefore 
also  presents  the  combination  of  the  quantities  by 
addition  as  far  as  the  sum  9.  Mdition  consists 
therefore  in  finding  a  quantity,  or  number,  equal  to 
two  or  more  other  quantities  taken  together '^  or,  a? 

2 


14  DOTATIONS   OF   THE   FOUR 

it  is  usually  called  :  to  find  the  sum  of  two  or  more 
numbers. 

§  15.  If  on  the  contrary  the  difference  of  two 
quantities  or  numbers  is  to  be  founds  the  operation 
is  called  subtraction.  In  this  operation  the  smaller  of 
two  numbers,  which  is  called  the  subtrahend,  is  taken 
away  from  the  larger  one,  and  the  result  is  called 
the  remainder ;  it  is  evidently  the  opposite  of  the 
foregoing.  Only  two  quantities  or  numbers  can  be 
concerned  in  a  subtraction,  for  one  result ;  if  more 
mimbers  are  to  be  subtracted,  it  must  be  done  by 
a  new  operation. 

§  16.  All  the  subsequent  operations  in  arithme- 
tic, are  combinations  of  the  two  preceding  ones  ac- 
cording to  certain  laws. 

§  17.  The  addition  of  the  same  number  or  quantity 
a  certain  number  of  times,  is  called  multiplication. 
When  this  is  treated  in  detail,  the  manner  in  which 
the  principles  of  multiplication  arc  deduced  from 
those  of  addition,  will  be  shown.  The  two  num- 
bers multiplied  into  each  other  are  called  factors, 
and  the  result  is  called  the  product. 

§  18.  The  opposite  of  the  operation  of  multiplica- 
tion^  is  called  division.  It  represents  a  successive 
.subtraction  of  the  same  number,  a  certain  number 
of  times,  from  another.  The  number  from  which 
this  successive  subtraction  is  made  is  called  the  divi- 
dend;  the  number  repeatedly  subtracted,  the  divi- 
sor; and  the  result,  the  quotient,  it  indicates 
how  many  times  the  divisor  is  contained  in,  or  can 
he  taken  away  from,  the  dividend. 

§  19.  These  four  operations  of  arithmetic,  addi- 
tionf  subtraction,  multiplication,  and  division,  are 
called  :  the  four  rules  of  arithmetic.  It  has  been  ob- 
served that  the  second  is  the  opposite  (»r  the  first, 
and  the  fourth  the  opposite  of  the  thira  ;  and  such 
must  be  the  case  in  any  system  cf  combination  of 
quantity  that  can  be  devised.  In  aU  arithmetic,  it  is 


I 


RUIES    OF   ARITHMETIC.  15 

always  necessary,  that  both  the  direct  and  inverse 
operation  shall  be  devised  ;  and  directions  or  rules 
deduced  and  given  for  their  execution. 

§  20.  To  facilitate  the  expression  of  the  idea  of 
these  four  operations  or  rules  of  arithmetic,  certain 
signs  are  made  use  of,  to  indicate  them  in  an 
abridged  manner,  which  it  is  proper  and  very  use- 
ful to  understand ;  their  use  will  conduce  to  clear- 
ness in  the  expression  of  the  operations  of  arithmetic. 

To  denote  an  addition  the  sign  (-j-)  is  used,  as  for 
instance,  if  7  and  2  ai-e  to  be  added,  this  will  be  writ- 
ten, 7+2,  and  in  the  same  way  for  more  numbers. 

To  denote  a  subtraction,  the  sign  ( — )  is  used,  so 
for  instance,  to  indicate  that  from  the  number  7,  the 
number  2  is  to  be  subtracted,  this  will  be  written, 
7—2. 

To  indicate  a  multiplication  the  numbers  are  se- 
parated by  a  full  stop,  (.)  or  by  this  sign.  (  X?)  thus 
to  indicate  the  multiplication  of  7  by  4,  we  write  7.  4 
or  7  X  4.  If  two  or  more  quantities  already 
united  by  -f-  or  —  are  to  be  affected  by  the  mul- 
tiplication with  one  number,  these  quantities  are 
inclosed  in  (  )  and  the  multiplier  written  to  them,  in. 
the  same  manner  as  before  to  the  single  number,  for 
instance  {7  -{-  5)  X  13  is  the  sum  of  7  and  5  to  be- 
multiplied  by  13. 

To  indicate  a  division,  two  different  signs  are  also 
made  use  of ;  either  by  placing  two  dots,  or  the  i 
colon,  (:)  after  the  dividend,  and  writing  the  divi- 
sor after ;  or  by  writing  the  divisor  under  the  divi- 
dend separating  them  by  a  horizontal  line,  thus 
8:2  or  f  denotes  that  8  is  to  be  divided  by  2. 

Besides  these  four  signs  we  are  yet  in  need  of  a 
sign  to  express  the  equality  of  two  quantities ;  this 
is  done  (by  two  horizontal  parallel  lines)  thus,  =. 

These  signs  will  suffice  here,  for  other  forms  of 
calculation,  or  combination,  other  signs  are  made 
use  of  ^  but  it  will  be  much  easier  to  understand  their 


16  NOTATIONS    OF   THE    TOUR 

meaning  when  the  subject  itself  is  treated ;  it  is 
therefore  more  proper  to  postpone  their  explanation 
for  the  present. 

§  21.  As  it  will  be  proper  for  the  scholar  to  ex- 
ercise himself  in  the  expression  of  these  signs,  in 
order  that  he  may  become  familiar  with  their  im- 
port, and  acquire  clear  ideas  of  arithmetical  opera- 
tions, I  shall  here  join  a  few  examples  of  the  four 
rules  of  arithmetic,  which  the  teacher  may  after- 
wards multiply. 

In  Addition.  7-f9=16,  means  the  addition 
of  seven  and  nine  is  equal  to 
16,  or  the  sum  of  7,  and  9,  is  16. 
7-f3  +  8  =  18;or  the  addition 
of  seven  and  three  and  eight  is 
equal  to  18,  or  the  sum  of  7,  and 
3,  and  8,  is  18. 
In  Suh traction.  13  —  7  =  6,  means  the  dif- 
ference between  13,  and  7,  is 
equal  to  65  or  7  taken  from 
13,  leaves  6;  for  example:  we 
shall  have,  joining  both  the  pre- 
ceding notations,  1 3  -|-  9  —  8  = 
22  —  8  =  14  5  which  as  is  shown 
by  the  above  example^  it  will  be  easy 
for  any  scholar  to  express  in  words, 
but  the  idea  conceived  as  it  is  writ- 
ten by  signs  is  the  best  mode  of 
expressing  it. 
In  Multiplication.  As  a  repeated  addition  of  the 
same  number  first,  then  as  a  mul- 
tiplication of  factors,  equal  to  a 
certain  product,  it  will  be  expressed 
as  in  tlie  following  examples : 

74.74-7-1-7  =  4.  7  =  4   X7  =  28. 
54-54-54-5-1-5  =  5. 6  =  5x  5  =  25. 
and  thus  in  any  other  case. 


RULES   OF   ARITHMETIC.  If 

In  Division,  If  we  express  division  by  a  suc- 
cessive subtraction  of  one  number 
a  certain  number  of  times  from, 
another,  we  shall,  in  the  case 
of  this  subtraction  exhausting  the 
number,  reduce  it  to  0,  and  there- 
by show  that  tlie  divisor  is  con- 
tained in  the  dividend,  exactly  as 
many  times  as  it  has  been  possible 
to  subtract  it,  so  we  would  have 
for  instance, 

3a  —  6  --  6  —  6  —  6  —  6  =  0. 

Which  showing  that  six  subtracted  five  times  from 
30,  and  its  successive  remainders,  leaves  nothing  | 
therefore,  if  we  express  this  as  a  division,  having 
the  result,  or  quotient,  on  the  other  side  of  the  sign 
of  equality  we  obtain  in  this  case  the  expression 
30  :  6  =  3_o  =  5 

If  the  successive  subtraction  of  the  divisor,  should 
at  last  leave  a  number  smaller  than  this  divisor,  it 
will  give  what  is  called  a  remainder,  that  is  still 
affected  with  the  sign  of  division  by  the  divisor ;  as 
for  instance  in  the  following  example : 

36  —  8--8--8  —  8  =  %6=:4^i. 

This  last  part  of  the  expression  indicates  a  division 
that  can  no  longer  be  executed,  on  account  of  the 
divisor  being  greater  than  the  dividend ;  it  no 
longer  gives  a  whole  quantity  in  the  result;  these 
expressions  are  called  fractions,  and  we  thus  have 
already  the  fundamental  idea  of  a  fraction,  from 
which  we  shall  hereafter  deduce  the  principles  of 
calculation  that  are  adapted  to  them. 

§  22.  By  means  of  these  explanations  of  the  prin- 
ciples, and  the  notations,  of  arithmetic,  it  is  proper 
for  the  teacher  to  introduce  his  scholars  to  the  sub- 
iect,  and  prepare  them  for  its  future  practical  appli- 
2  ^ 


18  ADDITION-. 

cation,  if  he  would  not  make  it  a  study  toilsome  to 
the  boy,  and  an  equally  toilsome  task  for  himself. 
No  teacher  ever  had  a  scholar  who  did  not  ask  him 
{why 'I)  when  he  directed  him  to  do  something;  and 
•this  why,  the  reasonable  and  faithful  teacher  must 
answer  in  a  satisfactory  manner ;  this  will  be  ren- 
dered easy  by  the  preceding  process,  elucidating 
the  principles  of  arithmetic.  The  reasoning  of 
the  child  must  be  cultivated,  if  he  is  ever  actually  to 
understand  arithmetic,  and  not  forget  it  when  out  of 
school,  or  out  of  practice,  as  will  be  the  case,  if  he 
has  only  committed  to  memory  dead  rules  for  which 
he  saw  no  reason.  By  such  a  process  arithmetic 
will  ever  be  agreeable  to  the  scholar,  as  an  exercise 
of  his  intellect  within  the  limits  of  his  capacity. 
The  time  spent  in  explaining  and  reasoning  with  the 
scholar  upon  these  principles  will  be  amply  gained 
by  his  more  successful  and  regular  progress  in 
arithmetic,  when  applying  it  to  each  individual  rule 
and  case. 


CHAPTER  III. 

The  four  rules  of  Arithmetic,  in  whole  numbers. 

•  §  23.  ADDITION,  has  been  defined  as  the  me- 
thod of  finding  a  quantity,  equal  to  two  or  more 
quantities,  taken  together.  Its  expression  as  a 
problem,  is  therefore  :  to  find  the  sum  of  two  or 
inore  quantities.  /From  what  has  been  said  of  the 
principles  of  the  system  of  numeration,  in  common 
arithmetic,  it  follows  :  that  in  order  to  prepare  the 
given  numbers  for  addition,  they  must  be  written 
under  each  other  so  as  to  bring  the  units  of  the  one 
under  the  units  of  the  other ;  and  so  all  the  numbers 
successively  higher  in  the  order  of  the  system  of 
uumeration^  will  each  come  under  its  equal  deno- 


ADDITION.  19 

mination ;  by  which  means  they  may  be  added  the 
more  easily. 

Then  the  numbers  are  added  together,  in  this 
order,  beginning  always  with  the  unit  and  proceed^ 
ing  until  we  reach  the  last  on  the  left  hand  side. 

Example.— To  add  176873  +  34719. 


Write  these  numbers  thus  ; 

176873 

34719 

and  draw  a  line  beneath  them ; 
then  add  the  column  of  units, 

12 

tens, 

8 

hundreds, 

16 

thousands, 

10 

ten  thousands, 

10 

hundreds  of  thousands, 

1 

211692 

placing  each  particular  sum  so  that  the  figure  on  the 
right  hand  shall  be  under  the  numbers  added;  then 
draw  a  line  and  add  the  numbers  as  they  are  now  pla- 
ced. The  result  thus  obtained  will  be  the  sum  of  the 
numbers  added.  It  is  evident,  here,  that  whenever 
the  sum  of  any  one  of  these  individual  additions 
exceeds  what  can,  in  our  system  of  notation,  be 
written  with  a  single  figure,  we  had  to  place  the 
figure  coming  to  the  left  of  it,  under  the  next  high- 
er order  ;  and  in  the  second  addition,  these  numbers 
were  then  added  to  the  result  of  the  addition  next 
following.  This  can  therefore  he  done  at  once,  by 
the  following  process. 

Having,  as  in  the  example,  found  the  first  sum, 
9  and  3,  which  is  12,  (or  9  -f-  3  =  12)  the  2  is  placed 
under  the  unit,  and  the  1  is  kept  in  memory  to  be  add- 
ed to  the  next  operation,  in  this  case  to  the  sum  of  the 
tens,  (which  is  called  carrying  ;)  so  that  in  this  next 
addition  you  say  ;  7  +  1  +  1  =  9  or  7  and  1  is  8, 
(as  marked  in  the  example,)  and  1  carried  gives  9, 
which  is  immediately  written  to  the  left  of  the  former 


20  ADDITION. 

result,  or  under  the  tens ;  this  number  can  be  writ- 
ten entirely,  and  therefore  gives  nothing  to  carry. 
The  next  or  hundreds  would -give  8  -f  7  =-  15,  or 
8  and  7  is  15 ;  write  5  and  keep  1  ;  then  the  next, 
6  and  4  is  1 0,  and  1  kept  is  II;  (or  6  +  4  -f  1  =  U ;) 
and  so  on  to  the  last  figure  on  the  left  hand. 

§  24.  If  there  is  a  greater  number  of  figures  to 
be  added  the  same  mosle  of  operation  is  used,  only 
repeated  as  often  as  the  number  of  figures  given 
will  require ;  as  for  instance  in  the  following  exam- 
ple: 

To  find  the  sum  of  674-21  -}-  389  -f  641827  -f  30 
-f-  4  +  7259  = 

Write  these  numbers  all  under  each  other  so  that 
the  units  fall  in  the  same  column,  and  the  other 
numbers  successively  under  their  respective  places, 
thus  : 

67421 

389 

641827 

30 

4 

7259 


71(3930 


Then,  having  drawn  a  line  beneath,  begin  again  by 
saying,  in  the  column  of  the  units,  9  +  4  -}-  0  -h 
7  -f  9  +  1  =  30,  write  0,  and  keep  3;  then  for  the 
second  column,  or  that  of  the  tens,  say  :  3  -f  5  +  3  -j- 
2  +  8  +  2  =  23;  write  three  and  keep  two,  and 
proceding  in  this  manner  to  the  last  figure  on  the 
left  hand;  which  will  produce  the  sum  found  in  the 
example,  under  the  line.  It  is  necessary  to  practice 
such  examples  sufficiently,  until  the  scholar  can  exe- 
cute them  with  facility  and  accuracy,  so  that  it 
becomes  to  him  an  easy  mechanical  practice.  It  is 
proper  to  mix  the  numbers  of  different  orders,  as 
above  at  once  and  not  to  distinguish  separate  cases, 


SUBTRACTION^.  Q{ 

iu  order  that  the  scholar  may  seize  the  principles 
of  the  operation  intellectually,  and  with  reflection, 
and  not  by  mere  memory  and  habit. 

§  25.  SUBTRACTION,  as  has  been  already 
said,  is  the  opposite  of  addition  ;  its  Problem  is : 
to  find  the  difference  between  two  numbers. 

In  common  arithmetic  it  is  always  required,  that 
the  number  to  be  subtracted  be  greater  than  the 
number  from  which  it  is  to  be  subtracted  ;  otherwise 
the  result  would  become,  what  in  universal  arithme- 
tic is  called  negative  :  that  is  to  say  in  denying  the 
possibility  of  the  subtraction  it  would  indicate  the 
number  from  which  it  was  intended  to  be  subtracted 
to  be  so  much  too  small  to  admit  this  subtraction, 
as  the  number  found  indicates. 

This  operation  is  necessarily  limited  to  two  num- 
bers or  quantities,  if  more  should  be  concerned  in  a 
question,  the  result  must  be  obtained  by  a  repetition 
of  the  operation. 

§  26.  Of  this  operation  in  simple  numbers  we 
have  given  the  principle  in  the  explanation  of  the 
signs,  as  in  the  case  of  addition  ;  when  the  numbers 
are  larger  the  following  is  the  preparation  and  the 
operation. 

Write  the  number  from  which  the  subtraction  is 
to  be  made  first,  and  the  subtrahend  under  it,  in 
such  a  manner  that  the  unit  comes  under  the  unit, 
and  the  following  numbers,  to  the  left,  each  under 
its  similar  superior  number,  and  draw  a  line  undei- 
them  thus ; 

9643187 

7532043  =  Subtrahend, 


2111144  =  Remainder, 


9643187  =  Proof, 
then  take  the  difference  between  each  of  the  corres- 
;ponding  numbers,  beginning  by  the  unit,  and  write 


r22  STTBTRACTIOSr. 

the  difference  directly  under  these  numbers,  the 
number  resulting  therefrom  will  be  the  entire  dif- 
ference between  the  two  given  numbers. 

As  well  from  the  principle  that  this  operation  is  the 
opposite  of  the  addition,  as  from  the  consideration  of 
the  preceding  operation,  it  may  easily  be  observed  : 
that  the  proof  of  the  correct  execution  of  this  opera- 
tion may  be  given,  by  adding  the  result,  or  remainder 
obtained,  to  the  lower  number  above  the  line,  or  the 
subtrahend,  wliich  addition  must  give  the  first  or 
npper  number  for  its  result.  It  is  therefore  proper 
to  accustom  beginners  to  make  this  proof,  in  order 
that  they  may  have  the  satisfaction  of  verifying  the 
correctness  of  their  operation  ;  drawing  therefore  a 
line  under  the  result,  the  two  numbers  immediately 
above  are  added,  when  the  first  number  must  again 
appear  in  the  result. 

§  £7.  In  this  operation  it  may  evidently  occur  : 
that,  though  the  quantity  from  which  another  is 
to  be  subtracted  may  be  greater,  some  of  the  in- 
dividual numbers,  of  the  inferior  order,  in  the  sub- 
trahend ,•  may  be  larger  than  those  corresponding 
to  them  in  the  superior  number. 

In  this  case  it  becomes  necessary  to  supply  the 
want  by  borrowing  an  unit  from  the  next  higher  or- 
der of  the  upper  number,  which  will  of  course  then 
represent  a  ten  in  its  corresponding  order  next  infe- 
rior in  place  and  value,  and  furnishing  of  course  al- 
ways in  addition  to  this  number  itself  a  larger  number 
than  that  in  the  subtrahend,  will  admit  the  latter  to 
betaken  from  it,*  the  remainder  is  then  written  in  its 
proper  place,  and  if  even  tlie  preceding  superior  num- 
ber were  an  0,  the  lending  being  considered  as  possi- 
ble from  the  preceding  higher  order,  the  operation 
would  be  the  same,  an  unit  would  be  borrowed  from 
it,  and  the  number  afterwards  called  9,  again  under 
the  supposition  before  made  of  the  lending  being  made 
from  the  next  higher  order,  which,  when  reached. 


SUBTRACTIOBT.  2S 

is  considered  as  diminished  by  an  unit.  It  is  evi- 
dent that  if  the  superior  number  is  larger  than  the 
inferior  or  subti^ahend,  this  lending  will  always  be 
compensated  before  the  end  of  the  operation,  what- 
ever be  its  extent,  through  the  figures  preceding  the 
last  on  the  left  hand  side. 

Let  the  following  example  be  given. 
600198056  —  336499278 

Place  the  example  as  indicated,  thus  :\ 

600198056 
356499278 


243698778 


600198059 
Here  in  the  units  the  8  cannot  be  taken  from  the  6, 
an  unit  is  therefore  borrowed  from  the  5  in  the  tens 
preceding  the  6,  which  added  to  the  6,  gives  16, 
from  which  the  8,  being  taken  leaves  8,  to  be  writ- 
ten in  the  place  of  the  units.  (For  beginners  it  will 
be  proper  to  mark  every  figure  from  which  an  unit 
has  thus  been  borro\Ved,  by  a  dot  above  it,  which  is 
done  in  order  that  it  may  not  be  forgotten  to  pay 
attention  to  it  in  proper  time.) 

In  the  second  place  or  the  tens  we  have  then 
only  a  4,  instead  of  a  5  ;  we  are  therefore  again  un- 
der tlie  necessity  of  borrowing  from  the  next  higher 
figure,  though  this  be  an  0,  subtracting  then  7,  from 
14,  the  remainder  7,  is  written  in  the  proper  plac 
In  the  place  of  the  hundreds  we  have  then,  by  the 
effect  of  the  foregoing  borrowing,  which  is  trans- 
ferred to  the  place  of  the  thousands  a  9,  from  which 
the  2  subtracted  gives  the  remainder  7.  By  the 
preceding  borrowing,  the  8,  in  the  order  of  the  thou- 
sands has  now  become  a  7,  and  is  again  insufficient 
to  admit  of  a  9  being  subtracted  from  it ;  the  borrow- 
ing of  an  unit  of  the  higher  order  gives  here  17. 


S4  SUBTRACTION. 

from  wliich  9  being  taken  gives  8,  as  remainder.  The 
9  in  the  next  higher  order  has  now,  by  the  lending 
become  an  8,  in  order  to  subtract  the  9  below,  from 
it,  a  unit  of  the  next  higher  order  is  again  borrowed, 
making  it  18,  subtracting  9  from  it,  gives  9,  as  the 
remainder  to  be  written.  The  unit  in  the  next 
higher  order  having  been  borrowed,  the  0  remain- 
ing, is  made  into  a  10,  by  borrowing  an  unit  from 
the  next  higher  order,  from  which  4  being  sub- 
tracted, leaves  6 ;  the  next  higher  number  being 
a  9  by  the  supposed  borrowing  from  the  higher  or- 
der, and  the  same  being  the  case  for  the  next  fol- 
lowing 0,  these  two  subtractions  are  made  exactly 
like  that  in  the  hundreds,  until  ultimately  the  last 
left  hand  figure  being  higher  than  the  number  of  the 
subtrahend  under  it,  the  subtraction  is  possible 
"which  being  done,  the  number  243698778  presents 
the  full  remainder  required  by  the  subtraction,  or 
is  the  difference  between  the  two  given  numbers. 

The  proof  of  the  correctness  of  this  operation  will 
again  be  found  by  the  addition  of  the  subtrahend 
and  the  remainder,  which  by  carryings  correspond- 
ing to  the  preceding  borrowing,  will  again  give  the 
upper  immber,  as  seen  by  the  example.  Proper 
attention  to  the  example  here  explained  will  teach 
bow  to  act  in  every  case  tliat  may  occur  in  subtrac- 
tion, and  it  will  be  proper  for  the  scholar  to  be 
exercised  upon  a  sufficient  number  of  examples,  that 
he  may  acquire  facility  in  this  operation. 

§  28.  There  are  two  otlier  ways  to  perform  the 
operation  to  obtain  the  same  result;  but  the  above 
explained  course  of  reasoning  is  the  one  most 
closely  connected  with  tlie  nature  of  the  question, 
and  the  implied  requisites  of  the  operation ;  it  is 
therefore  proper  to  keep  the  scholar  to  this  con- 
sideration. When  once  he  has  gone  through 
the  whole  course  of  arithmetic  he  will  easily  see 
the  two  other  methods,  which  if  taught  at  this  stage 


MFI.TIPI.1CATI0N.  25 

gf  the  study  would  contuse  his  ideas^  and  are  there- 
tore  intentionally  omitted  here. 

$  29.  MULTIPLICATION,  as  has  been  stated, 
is  the  addition  of  a  given  number  repeated  as  many 
times  as  another  number  contains  units,  or  indi- 
cates; thus  every  number  is  in  itself  the  pro- 
duct of  that  number  into  the  unit.  It  is  indiiferent 
which  of  the  two  numbers  be  considered  as  acting 
the  one  or  the  other  part  in  the  operation ;  therefore 
they  are  both  equally  called  Factors  ;  the  result  of 
the  operation  is  called  the  product. 

It  is  necessary,  in  order  to  perform  this  operation 
with  ease,  in  more  complicated  calculations,  to  com- 
mit to  memory  the  product  of  the  nine  numbers  ex- 
pressed by  our  numerical  symbols.  It  is  needles3 
for  written  operations  to  go  any  farther,  because  the 
higher  multiplications  overreach,  in  writing,  oxsy 
system  of  numeration. 

We  have  already  seen  that  our  system  of  numera- 
tion is  a  successive  addition  of  the  unit  below  9, 
which  being  the  last  symbol  of  quantity,  the  next 
quantity  is  expressed  by  a  change  of  place.  If  now 
we  treat  every  one  of  the  nine  symbols  in  the  same 
way,  by  the  successive  addition  of  itself,  we  obtain^ 
successively,  the  product  of  each  of  these  symbols  in. 
a  similar  manner,  forming  what  is  commonly  called 
the  multiplication  table.  Writing  therefore  the  re- 
gular series  of  numbers  as  far  as  9,  in  a  horizontal 
line,  add  each  of  them  to  itself,  writing  the  result 
under  it,  then  to  this  sum  adding  again  the  number^ 
and  so  in  succession,  until  the  whole  9  symbols  are 
exhausted,  we  shall  have  the  following  system  of  re- 
sults : 


9.6 


MriTIPMCATlON'. 


'<><>0<>0'<><><><><>C<><>i.><><><><^^ 


8 
10 


12 


12 


16 


15      20 


><>C<>000<>0-0<><><>< 


4^ 


10 

12 

1^ 

18 

20 

24 
30 

25 

30 

36 

35 

42 

40 

48 

54 


14 


21 


28 


35 


42 


49 


56 


63 


•C:.<»<>0<'-0-0-0-00< 


Considering  the  preceding  table,  we  find  that  the 
first  column  to  the  left,  which  again  contains  the  se- 
ries of  natural  numbers  of  our  system  of  symbols,  by 
the  successive  addition  of  the  unit,  keeps  an  account 
of  all  the  other  successive  additions ;  or  that  it  indi- 
cates how  many  times  this  addition  has  been  repeat- 
ed, and  that  the  result  of  any  number  of  such  addi- 
tions, of  any  one  of  the  successive  numbers,  is  al- 
ways found  in  the  meeting  of  the  horizontal  and 
vertical  lines  of  the  two  numbers  taken  as  factors ; 
thus,  for  instance,  under  7,  and  where  the  horizontal 
line  marked  6,  in  the  first  column,  meets  it,  we  find 
42,  that  is,  the  addition  of  7  six  times  repeated  gives 
tlie  result  42.  In  like  manner  under  6,  opposite  to 
the  7  in  the  first  column,  will  again  be  found  42. 
So  6  times  7,  and  7  times  6,  (such  is  the  usual  ex- 
pression,) are  equivalent ;  as  has  been  stated  above  t 
and  such  is  the  case  with  any  other  number. 


MULTIPLICATION.  27 

The  regular  progression  of  the  different  results 
is  easily  observable,  and  some  attention  to  it  will 
assist  in  fixing  them  in  the  memory ;  it  is  best 
not  to  load  the  beginner  with  a  longer  table,  for 
which  he  has  no  use,  until  he  may,  in  practical  ap- 
plication, wish  to  calculate  from  memory,  without 
writing,'  when  the  circumstance  of  its  possessing 
interest  and  usefulness  will  make  that  task  easy, 
which  at  this  stage  of  instruction  is  a  dry  and  use- 
less labour. 

§  30.  We  must  now  suppose :  that  the  scholar  has 
acquired  some  facility  in  the  use  and  application  of 
the  results  of  the  preceding  table  ;  and  shall  proceed 
to  show  the  details  of  multiplication  by  examples. 

Be  it  given  to  multiply  357279  by  6  ;  or  to  exe- 
cute what  is  expressed  by  the  sign  of  multiplication, 
thus:   6  X  357279. 

Write  the  smaller  factor,  in  this  case  the  6,  under 
the  other,  so  that  the  units  stand  under  each  other  ; 
then  execute  the  multiplication  of  each  of  the  num- 
bers of  the  larger  factor  successively,  and  write 
the  result  under  the  horizontal  line  drawn  below 
the  factors,  so  that  the  right  hand  figure  of  the  pro- 
duct shall  always  stand  under  the  number  multi- 
plied, thus : 

367279 
6 


64 
42 
12 

42 
30 
18 


2143674 
then  adding  up  all  these  products,  the  sum  result- 
ing will  be  the  general  product  of  the  whole  multi- 
|)lication. 


28  MrLTIPLICATION. 

The  inspection  of  this  detailed  execution  of  the 
preceding  example,  shows  that  we  may  again  apply, 
in  this  case,  the  mode  of  abridgement  that  has  been 
pointed  out  in  addition.  We  would,  therelbre,  in 
the  preceding  example  say,  (analogous  to  what  has 
been  done  in  addition,)  6  times  9  is  54  ;  write  4, 
and  keep  (or  carry)  5  ;  then  keeping  this  5  in  mind., 
we  would  next  say,  6  times  7,  is  42,  and  5,  is  47  ; 
writing  again  the  7,  and  keeping  the  4  to  be  added 
to  the  next  product;  then  6  times  2,  is  12.  and  4,  is  16 ; 
when  writing  the  6,  and  keeping  1,  and  proceeding 
thus  to  the  end  of  the  number,  we  obtain  at  once 
the  same  numbers  that  appear  above,  in  the  final  re- 
sult. This  mode  of  proceeding  is  therefore  the 
usual  mode  of  operating,  with  each  of  the  numbers 
of  the  factor  that  is  chosen,  for  the  purpose  of  taking 
the  multiples  of  the  other  by  it ;  for  which,  as  said 
before,  it  will  be  best  to  choose  the  smaller  one,  be- 
cause it  gives  the  shorter  example  in  writing. 

§  31.  But  when  both  factors  are  compound  num- 
bers, it  is  evident  that  the  multiplication  of  each  of 
the  numbers  of  the  one,  cannot  be  made  at  once  with 
all  the  numbers  of  the  other ;  therefore  we  must  pro- 
ceed with  each  number  of  the  one  factor,  exactly  as 
shown  above  with  the  single  number ;  and  in  order 
to  give  to  each  individual  result  its  proper  place,  we 
must  begin  to  write  the  first  number  of  each  pro- 
duct on  the  right  hand  side,  exactly  under  the  num- 
'ber  of  the  multiplier  of  which  it  is  the  product ;  as 
its  proper  unit.  The  sum  of  all  these  partial  products 
is  then  made,  by  the  addition  of  all  the  numbers  in  the 
regular  order  in  which  they  stand  under  each  other, 
as  this  has  been  done  in  the  preceding  example,  with 
the  partial  products  of  the  simple  number. 

This  shall  be  shown  in  the  following  example,  in 
which  it  is  required  to  perform  the  multiplication 
174392  X  6435;  writing  the  factors  properly  under 
each  other,  so  that  the  units  stand  under  each  other^ 
and  the  other  numbers  follow  in  their  regular  order. 


MULTIPIICATION-.  29 

the  successive  results  in  their  proper  places,  will  be 
as  follows  :  - 

174392 
6435 


871960 
523176 
697568 
1046352 

1122212520 
In  this  manner,  every  example,  whatever  quantit) 
of  figures  it  may  be  composed  of,  will  stand. 

If  any  of  the  figures  in  the  number  to  be  multi- 
plied, which  is  called  the  multiplicand,  should  be  an 
0,  its  product  into  any  number  whatsoever,  is  =  0  5 
because  0  times  any  number  whatever,  always  indi- 
cates that  the  number  is  not  there ;  the  place  will, 
therefore,  receive  only  that  number  which  may  be 
t^arried  over  from  the  preceding  multiplication,  and 
if  none  be  carried,  only  an  0. 

If  an  0,  occurr  among  the  numbers  by  which 
the  multiplication  is  to  be  performed,  or  the  multi- 
plier, the  whole  row  of  figures  to  be  multiplied  by 
it  producing  a  result  =  0,  the  place  where  the  first 
number  would  stand  will  only  be  marked  by  an  0^ 
and  the  multiplication  by  the  next  following  num- 
ber is  begun  in  the  same  row,  immediately  after, 
thus  placing  each  result  in  its  proper  place. 

The  following  example  will  explain  both  the  above 
cases,  where  the  effect  of  the  two  O's,  in  the  multi- 
plier is  shown  by  the  removal  towards  the  left  of 
the  two  latter  rows  of  figures. 

3603904 
50203 


10811712 
72078080 
180195200 

3  *  180926792512 


30  DIVISION-. 

§  32.  It  will  be  proper  to  ©xercise  the  scholar  in 
a  variety  of  examples,  until  he  has  become  accus- 
tomed to  the  operation,  and  is  able  to  make  any 
multiplication  without  error :  the  younger  the 
scholar  may  be,  the  easier  the  examples  must  be  in 
the  beginning,  and  must  gradually  increase  in  diffi- 
culty, by  the  combination  of  different  cases,  and  larg- 
er numbers.  Still,  in  this  it  is  to  be  observed  :  that 
when  the  beginner  has  performed  examples  gradu- 
ally with  the  whole  series  of  the  nine  simple  num- 
bers, it  will  be  proper  to  show  him  only,  what  is 
the  effect  of  a  compound  multiplier,  as  a  repetition 
of  the  similar  operation  of  one  number  only,  and 
the  addition  of  the  different  partial  products  into 
one  whole;  and  not  to  follow  servilely  the  augmenta- 
tion by  one  number,  (or  place  of  figures,)  that  he  may 
not,  as  often  happens,  consider  that  he  has  every 
time  a  new  difficulty  to  overcome,  but  must  himself 
come  to  the  observation,  that  multiplication  by  a  num 
ber  of  places  of  figures  is  a  mere  repetition  of  the 
operation  he  knows,  requiring  nothing  but  a  little 
more  attention,  and  more  accuracy  in  the  placing 
of  the  figures. 

§  33.  DIVISON,  is  an  operation  the  opposite  of 
Multiplication,  as  has  already  been  stated  ;  its  pro- 
blem is  therefore  :  to  find  how  many  times  a  givea 
number  is  contained  in  another  given  number,  which 
is  thus  considered  as  a  product  of  the  first  and  the 
quantity  sought. 

The  table  of  products,  or  multiplication  table, 
^iven  above,  may  therefore  be  here  applied  inverse- 
ly; a  ready  and  habitual  knowledge  of  its  results  is 
tlierefore  also  constantly  applied  in  this  rule,  by  the 
comparison  of  its  results  with  the  quantities  pre- 
senting themselves  in  an  example. 

While  all  the  preceding  operations  have  begun  at 
the  unit,  this  on  the  contrary  must  begin  by  the 
highest  number,  or  order  of  symbols;  for  the  greater 


BIVISIOX. 

number  of  times,  which  one  quantitj^  may  bfr  con- 
tained in  another  is  necessarily  to  be  taken  out,  or 
considered,  first,  the  inferior  numbers  will  then  fol- 
low in  their  regular  order,  and  keeping  account  of 
the  value  of  any  remainder  from  the  preceding  ope- 
ration in  its  proper  rank,  as  in  the  following  example, 
which  we  shall  express  in  the  manner  that  has  been 
shown  in  §  20,  in  order  to  accustom  the  learner  to 
keep  the  systematic  language  of  the  operation  itself, 
which  is  always  the  most  preferable  method  ;  with 
this  view  we  shall  draw  a  horizontal  line  under  the 
dividend,  under  which  we  shall  place  the  divisor,  and 
the  result,  or  quotient,  will  be  written  on  the  right 
hand  side  of  the  sign  of  equality  which  follows 
them,  thus : 

842316 

=  280772 

3  3 


842316 


24 
24 


23 
21 

21 


6 

6 

0 
Here  we  say  S,  in  8,  is  contained  twice,  and 
having  written  the  2,  as  the  first  number  to  the 
quotient,  we  must  make  the  product  of  it  by 
the  divisor,  write  it  under  the  corresponding  num- 
ber of  the  dividend,  and  subtract  it  from  it,*  this  pro- 
duct being  6,  in  this  case  the  subtraction  leaves  2, 
as  a  remainder.  Now,  for  tho  sake  of  easier  distijic- 


32  DIVISION. 

tion  we  place  the  next  number  by  the  side  of  this 
remainder,  which  being  4,  gives  for  the  next  num- 
ber to  be  divided  24.  Now  3,  is  in  £4,  contained  8 
times  ;  placing  the  8  in  the  quotient,  multiplying  the 
3  by  it,  the  product  of  3  times  8,  placed  under  the 
S4,  being  also  £4,  leaves  no  remainder ;  placing  the 
next  number  2  down,  we  find,  that  3  not  being  con- 
tained in  it,  we  must  indicate  this  by  an  0,  in  the 
quotient,  for  the  rank  or  order  of  the  numeric  sys- 
tem corresponding,  which  being  Jone,  tlie  next  num- 
ber, 3,  is  taken  down  to  the  right  side  of  the  2, 
which  making  23,  we  say  3  in  23  will  be  contained 
7  times  ;  writing  the  7  in  the  quotient,  multiplying 
the  3  by  it,  and  subtracting  tlie  product  21  from 
the  23,  we  obtain  the  remainder  2 ;  taking  down 
the  1  which  gives  21,  we  say  again,  3  in  21,  is  con- 
tained 7  times,  and  the  product  3  times  7  being 
equal  to  21,  leaves  iio  remainder;  lastly,  bringing 
down  the  6,  we  find  3  in  6  twice,  and  writing  the 
2  in  the  quotient,  and  subtracting  its  product  by  3, 
from  the  6,  we  obtain  the  exact  quotient  280772. 

Division  being  the  opposite  of  multiplication, 
we  have  tlie  means  of  proving  this  result,  by 
the  multiplication  of  the  quotient  by  the  divisor  : 
the  product  of  which  must  be  equal  to  the  dividend, 
as  is  evident  from  the  definitions  given  of  this  ope- 
ration. 

Writing  then  the  divisor  under  the  quotient,  and 
performing  the  multiplication,  the  product  resulting 
will  be  equal  to  the  dividend,  if  the  whole  operation 
has  been  rightly  performed. 
5  34.  If  the  divisor  is  not  contained  an  exact  whole 
number  of  times  in  the  dividend  there  will  remain 
at  the  end  of  the  division,  a  number  smaller  than  this 
divisor,  which  is  called  the  remainder.  In  order  to 
indicate  fully  the  actual  result  of  the  division,  this 
number  is  yet  to  be  placed  at  the  end  of  the  quotient, 
with  the  divisor  written  under  it,  and  it  horizontal 


Divisioif.  35 

line  between  them,  to  indicate  that  this  division 
should  yet  be  made. 

Such  numbers  as  indicate  a  division  which  can- 
not be  executed,  are  called  proper  fractions,  while 
every  division,  indicated  as  above,  of  a  number 
larger  than  the  divisor,  is,  in  comparison  with  these, 
oalled  an  improper  fraction  :  and,  when  considered 
in  this  point  of  view,  the  number  corresponding  to 
the  dividend,  is  called  the  numerator,  and  the  num- 
ber corresponding  to  the  divisor  is  called  the  de- 
nominator; while  the  quotient,  whatever  it  maybe, 
will  always  represent  the  value  of  the  fraction. 

This  general  idea  of  fractions,  the  origin  of  which 
it  is  proper  to  show  here,  will  hereafter  be  the  fun- 
damental idea  from  which  the  calculation  of  this 
kind  of  quantities  is  to  be  deduced. 

The  following  is  an  example  that  will  show  such  a 
division,  and  the  mode  of  operating  in  the  case. 

Being  given  to  divide 


7835921 

=  979490  l- 
0 

8 
72 

63 

66 

7835921 

75 

72 

39 
32 

72 
71 

01 


In  this  example  :  we  see  that  thejfirst  number pf  the 
highest  order  being  smaller  than  the  divisor^  wo 


54  DIVISIOK-. 

must  take  it  jointly  with  the  next  following  lower 
numher  and  say  :  8,  in  78,  is  contained  9  times ;  and 
the  9  is  written  as  the  first  numher  in  the  quotient ; 
then  making  the  product  8.9  =  72,  and  writing  ii 
under  the  78,  from  which  it  is  subtracted,  and  leaves 
6,  which  being  written  below  the  line,  and  the  next 
lower  number  3,  written  down  to  it,  gives  63  for  the 
next  number,  to  be  divided  by  8,  which  being  con- 
tained 7  times  in  it,  7  being  written  in  the  quotient^ 
the  product  7  X  8  =  56  is  written  under  the  63,  the 
subtraction  performed,  and  the  5  or  next  following 
number  placed  down  to  the  7  that  remains  from  the 
subtraction  ;  the  operation  is  thus  continued,  exactly 
as  in  the  former  example,  until  when  the  last  num- 
ber, 1,  is^set  down  at  the  side  of  the  0,  we  find  that 
8  is  no  longer  contained  in  1,  and  therefore  write 
an  0  in  the  quotient,  and  having  no  more  numbers 
in  the  dividend,  we  find  that  1  ought  yet  to  be  di- 
Tidedby  8,  which  we  write  in  the  quotient,  as  stated 
above,  |  like  an  unexecuted  division,  or  a  proper 
fraction. 

When  we  make  the  proof  of  this  example,  as  has 
been  done  in  the  preceding  one,  we  consider  the  1  as  a 
remainder,  and  in  the  multiplication  of  the  quotient 
by  the  divisor,  add  it  to  the  product ;  so  that  we 
would  here  say  8  times  0,  is  0,  and  the  remainder 
1  added,  gives  1  for  the  first  number  of  the  product, 
exactly  as  in  the  dividend,  and  then  continue  the 
multiplication  through  the  whole  quotient  obtained, 
as  in  the  former  example. 

§  ^5,  When  the  divisor  is  a  number  composed  of 
more  than  one  figure,  the  principles  of  the  opera- 
tion remain  the  same  ;  but  it  becomes  necessary  to 
pay  attention  to  the  effect  of  the  multiplication  of 
the  quotient  into  the  whole  number  of  the  divisor : 
which  may  render  it  necessary  to  take  this  quotient 
smaller  than  might  appear  from  a  mere  compari- 
son of  the  first  numbers  of  the  divisor  and  the  divi- 


DIVISION.  3^5 

dend  ',  all  the  rest  of  the  operation  is  only  an  exten- 
sion of  the  operations  explained  in  the  preceding  ex- 
amples, which  have  heen  described  in  detail,  with 
the  express  view  of  giving  a  full  explanation  of  the 
first  elementary  principles.  Reasoning  with  the 
same  details  upon  the  following  example,  the  opera- 
tion of  a  division,  with  a  divisor  composed  of  more 
than  one  figure,  will  also  be  clear. 
The  following  division  being  given 


64U59213 


=   84510  ^^ 


758                   758 
G064  


34 1 9  422550 

3032  591570 


676080 
.50 
0 
633 


3872 

3790       64059213 


821 
758 


633 

Here  in  considering  only  the  first  number  of  the 
divisor,  and  comparing  it  with  the  two  first  of  the 
dividend,  we  would  find  7  in  64  contained  9  times; 
but  we  must  take  into  consideration  the  multiples 
of  the  numbers  which  follow  the  hundreds.  The 
5  tens,  or  50,  multiplied  by  9  would  give  45  tens, 
or  450,  and  7  X  9  =  63  would  leave  only  1,  which, 
considered  as  hundreds,  as  must  be  done  in  this 
case,  would  not  allow  us  to  take  the  4  hundreds; 
from  it.  We  find,  therefore,  that  the  quotient  9, 
is  too  large.  Taking  8,  we  find  that  7  X  8  =  56, 
leaves  8  as  remainder;  and  if  we  consider  now^the 
58,  as  multiplied  by  8,  we  find  that  the  4,  which 
roracs  here  again  as  hundreds  to  be  snhtnacted  fr^na 


^6  BIVISIOS-. 

the    8,  eau    be  taken  away  with   a   considerable 
remainder.     Writing   then    8,    as  the  first    num- 
ber in  the  quotient,  we  make  tlie  product  8  X  758, 
and  place  it  under  the  respective  numbers  of  the  di- 
vidend, so  that  the  product  of  the  first  number  of 
the  divisor,  that  is  to  say,  7X8  may  stand  under 
64,  and  the  otliers  follow  in  their  regular  order ; 
we  now  make  the  subtraction,  in  the  same  manner 
as  has  been  often  before  shown,  which  leaves  34 1  as 
remainder,*  as  this  is  less  than  the  divisor  it  also 
proves  that  no  greater  number  could  have  been  taken 
for  the  quotient ;  to  this  we  join,  as  in  the  preceding 
examples,  the  number  of  the  dividend  next  after  those 
used  in  the  last  subtraction,  which  is  here  9 ;  and 
now  proceed  as  before  to  compare  the  products  of 
7  with  the  34,  as  the  number  presenting  itself  here 
for  division,  in  the  same  rank  as  the  7  of  the  divi- 
sor ;  this  shows  4  as  the  nearest  factor  producing 
with  7  a  multiple,  (28,)  inferior  to  34,  and  leaving 
6  as  remainder,  while  4X58  giving  only  a  2  to  carry 
to  the  place  of  the  hundreds,  leaves  sufficient  room 
for  the  whole  product ;  we  thus  obtain  the  remain- 
der 387,  that  is  again  smaller  than  the  divisor,  and 
placing  after  it,  the  next  following  number,  2,  we 
say  first  7  into   38  is  contained  5  times,  and  the 
product,  5  X  7  ==  35,  taken  from  38  leaving  3,  the 
product  58  X  5,  giving  only  2  to  carry  to  the  place 
of  the  hundreds,  will  leave  a  sufficient  quantity  for 
the  subtraction ;  this  being  performed,  and  the  5 
placed  in  the  quotient,  we  have  the  remainder  82 ; 
then  placing  the  1  down  after  it,  the  resulting  821 
contains  the  divisor,  evidently,  only  once.   Placing 
1  in  the  quotient,  the  subtraction  leaves  73 ;  when 
the  last  figure,  or  3,  is  written  after  tliis,  the  num- 
ber 733,  that  results,  being  less  than  758,  the  lat- 
ter will  not  be  contained  in  it;  this  gives  an  0  in  the 
quotient,  for  the  last  whole  number ;  and  the  unexe- 
eutable  division  ^^  as  a  fraction  or  remainder,  as 
in  the  second  or  foregoing  example.  - 


DIVISION.  ST 

The  proof  of  this  example  is  again  made  in  the 
same  manner  as  in  the  last ;  multiplying  8450  x  758 
and  adding  733  to  it,  the  dividend  will  again  be 
obtained,  as  seen  in  the  example. 

The  remark  which  has  been  already  made,  upon  the 
propriety  of  practising  any  of  the  elementary  opera- 
tions until  a  competent  dexterity  is  acquired,  of 
course,  also  applies  here. 

The  detailed  manner  shown  hi  re,  is  what  is  usu- 
ally called  long  division  ;  and  even  experienced  cal- 
culators may  often  find  it  proper  to  apply  it,  when 
the  number  of  places  of  figures  in  the  divisor  is 
great. 

§  36.  For  common  calculation  it  is  often  desired 
to  spare  writing  out  the  numbers  for  the  subtrac- 
tion, and  writing  only  the  remainders.  This  is 
carried  on  as  in  the  follovving  example. 

Given  9460753 

=  10763/^V 

879  879 

6707         

6545  96867 

2713     75341 
76  8610476 


9460753 


Here  the  divisor  is  contained  once  in  the  three 
first  numbers  of  the  dividend  ;  the  1  being  placed  in 
the  quotient,  the  subtraction  is  immediately  made 
from  them,  and  only  the  remainder  placed  below  ; 
which  being  67,  and  the  next  number,  the  0,  being 
put  down  to  it,  the  divisor,  879,  being  larger  than 
670,  the  next  number  in  the  quotient  becomes  a  0. 
After  w  riting  it,  the  next  number,  7,  is  taken  down 
from  the  dividend,  and  in  the  resulting  6707  the 
divisor  is  contained  7  times.  Now  the  divisor  is 
multiplied  by  this,  and  the  subtraction  of  the  result 
made  in  the  memory  immediately,  and  again  only 
4 


38  Divisioir. 

the  remainder  wintteii  down,  thus :  say  7  times  9 
is  63,  subtracted  from  67^  which  the  number  above 
must  be  supposed  to  reprrsent,  in  order  to  allow 
the  subtraction  of  the  product  of  the  unit  or  first 
number,  leaves  4,  which  is  written  down  as  a  re- 
mainder under  the  7,  and  the  6,  which  the  num- 
ber in  the  next  higher  rank  has  been  supposed,  is 
kept  in  memory,  ajid  added  to  the  next  product  of 
the  tens,  or  next  higher  order  of  numbers,  with 
which  it  is  then  again  subtracted  ;  therefore,  con- 
tinuing the  multiplication,  we  say :  7  times  7  is  49, 
the  6  kept  being  a(hled  makes  55,  that  subtracted 
from  60,  which  we  suppose  to  bi-  the  number  above, 
having  the  0  in  the  first  place  to  the  right,  the  re- 
mainder, 5,  is  written  under  the  0,  and  6  is  kept  to 
add  to  the  next  following  product;  for  which  we 
say  7  times  8  is  56,  and  6  carried  is  62,  taken  trom 
67  leaves  5.  Bringing  now  the  5  from  the  dividend 
down  to  the  remainder  554,  we  have  for  our  next  di- 
vidend 5545,  in  which  we  say  :  8  in  55  is  contained  6 
times ;  and  as  6  X  8  =  48,  leaves  7  in  the  place  of 
the  hundreds,  for  the  carrying  of  the  lower  numbers 
following  it,  is  evidently  small  enough  to  allow  the 
subtraction  of  the  whole  product;  so  w^e  say  again, 
6  X  6  =  54,  from  55,  leaves  1  ;  write  it,  and  carry 
5 ;  then  6  X  7  =  42,  and  5,  is  47,  from  54,  leaves  7; 
write  7  down,  and  carry  5  ;  lastly,  6  X  8  =  48,  and 
5  is  53,  from  55,  leaves  2  ;  tlie  remainder,  presents 
therefore,  271 ;  to  which  the  3,  as  next  lower  num- 
ber in  the  dividend,  being  written,  we  find  7  in  27, 
is  contained  3  times,  or  3X7  =  21,  leaves  6,  a 
sufiicient  remainder  in  the  hundreds,  for  the  carry- 
ing of  the  product  3  X  79  ;  so  we  say  again,  3  X  9= 
27,  from  33,  leaves  6,  and  3  to  carry ;  then  3X7= 
21,  and  three  added  gives  24,  from  31,  leaves  7, 
and  3  to  carry ;  then  3  X  8=24,  and  3  is  27,  which  g 
subtracts  without  a  remainder,  from  the  27  above;  * 
and  the  remainder  76,  to  which  we  have  no  other 


TULGAR  rRACTIONS.  89 

number  to  set  down  from  the  divisor,  gives  the  nu- 
merator of  tlie  proper  fraction  remaining,  ■^\%^  as 
a  division  that  cannot  be  executed  with  our  present 
means.* 

In  the  manner  the  reasoning  has  been  carried,  in 
this  example,  every  other  more  complicated  case  is 
to  be  executed ;  it  is  therefore  expected  that  it  will 
suffice  to  introduce  into  the  practice  of  this  method. 


CHAPTER  III. 

Of  Vulgar  Fractions. 

§  37.  We  have  seen  already,  in  §  11,  and  at  the 
end  of  Division,  that  fractions  are  unexecuted  divi- 
sions 5  we  have  also  seen,  that  in  consequence  of 
this,  they  consist  of  two  parts,  corresponding  to 
the  two  parts  or  numbers  engaged  in  a  division : 
their  form,  or  the  manner  of  writing  them,  we  have 
seen  to  arise  naturally  fmm  the  division,  when  a 
number  remained  ultimately  in  the  dividend,  which 
was  smaller  than  the  divisor,  or  the  number  by 
which  it  should  be  divided  ;  we  have  there  already 
observed,  that  this  constituted  a  proper  fraction^ 
while  every  division  whatever,  exprcNsed  in  the 
same  form,  was  an  improper  fraction,  as  it  would 
naturally  be  called,  from  its  still  containing  the  di- 
visor a  whole  number  of  tijues. 

Tlie  number  above  the  horizontal  line,  (as  seen  in 
§  34,)  which  corresponds  to  the  dividend,  is  called 
the  numerator  of  the  fraction ;  and  the  number  be- 


*  The  proper  fractious^  are  still  i)urpoHely  here  repiesented  as 
unexerutable  divisions,  because  the  pre«"edins;  operations  Id  whole 
numbers,  do  not  furnish  any  mean-  for  such  a  divipion.  We  shall 
afterwards  show,  how  these  values  may  be  expressed,  either  ex- 
actly or  approximately,  by  a  continued  division,  and  an  extension 
of  the  decimal  system,  below  the  unit. 


40  VUXGA»  FRACTI0X9. 

low  this  line,  corresponding  to  the  divisor,  is  called 
the  denominator  of  the  fraction ;  thus  considering 
the  first  as  indicating  the  number  of  parts  taken, 
and  the  second  as  indicating  the  vajue  of  the  parts, 
or  giving  the  name  to  theparts.  By  this  means  any 
fraction  may  evidently  he  represented  as,  or  rather 
these  considerations  show  it  to  be  actually  the  pro- 
duct of  a  whole  number  into  unity,  divided  by  an- 
other number ;  and  this  latter  part  must  be  con- 
sidered as  characterizing  a  particular  kind  of  quan- 
tity, in  the  same  manner  as  the  different  places  of 
figur^^s  characterize  units,  tens,  hundreds,  and  so 
on  :  we  thus  evidently  have  (expressing  the  above 
reasoning  according  to  the  forms  and  signs  adopted) 
for  an  example, 

where  7  in  the  numerator,  counting  the  parts,  and 
18  the  denominator,  showing  these  parts  to  be 
eighteenths  of  the  unit.  And  the  value  of  these 
parts  may  evidently  be  as  much  varied  as  the  num- 
bers themselves ;  therefore  they  have  not,  like  the 
numerical  system,  one  necessary  and  uniform  law 
of  connexion. 

§  38.  From  these  considerations  of  the  principles 
and  nature  of  fraction,  the  following  three  funda- 
mental propositions  for  the  arithmetic  of  fractions, 
naturally  follow  : 

Proposition  i.  M  many  times  as  the  fiumerator  of 
a  fraction  is  made  larger  or  smaller^  the  denominator 
remaining  unchanged^  so  many  times  the  value  of  the 
fraction  is  made  larger  or  smaller. 

For,  by  multiplying  the  numerator  by  any  num- 
ber, there  are  as  many  times  more  parts  taken  as 
this  number  indicates,  and  in  dividing  it  by  any 
number,  there  are  as  many  times  less  parts  taken,  as 
the  number  indicates  ;  in  the  first  case,  therefore, 
the  value  of  the  fraction  is  as  many  times  larger,  and 


TUIiGAR  FRACTIONS.  41 

in  the  second,  as  many  times  smaller,  as  the  numher 
used  in  the  multiplication  or  division  indicates. 

13  X  7  7 

Example,     =  13  X  according    to    the 

18  18 

same  reasoDing  as  in  the  preceding  §. 

7:97 

And = :  9,  according  to  the  same. 

18  18 

Proposition  ii.  ^s  many  times  as  the  denominator 
of  a  fraction  is  made  larger  or  smaller,  the  numerator 
remaining  unchanged,  so  many  times  the  value  of  the 
fraction  is  made  smaller  or  larger. 

For,  the  ilc!Jominator  being  the  number  by  which 
the  unit  is  divided,  as  many  times  as  this  rmmber  is 
multiplied,  so  many  times  the  unit  is  divided  into 
more  parts ;  and  thereiorc,  the  parts  becoming  as 
many  times  smaller,  an  equal  number  of  them  re- 
presents a  value  as  many  times  smaller ;  that  is  to 
say,  the  value  of  the  fraction  is  as  many  times 
smaller,  and  inversely,  when  the  denominator  is  di- 
vided by  a  number,  the  unit  is  divided  by  a  number 
as  many  times  smaller  than  this  divisor  indicates  ^ 
therelore,  the  parts  become  as  many  times  larger, 
and  the  value  of  the  fraction  becomes  as  many  times 
larger ;  all  under  the  supposition  :  that  an  equal  num- 
ber of  these  parts  be  taken  before  and  after  the 
operation. 

7  7 

Example.   — is  13  times  smaller  thnn be- 

18  X  13  18 

cause  the  7  is  divided  by  a  number  1 3  times  larger  than  1 8  ; 

1 

r  we  have,    7  X 13  times  smaller  than 

18  X   13 

4  # 


42  rWLGAR  FRACTIONS. 

1  7  7 

7  X  ,  and =    —    is   9   times  larger  than 

18  18  :  9  2 

7 

,  because  the  7  is  divided  into  parts  9  times  larger  : 

18  ' 

1  1 

or,  we  have,  7  X  9  times  smaller  than  7  x  — 

18  2 

Proposition  hi.  When  the  numerator  and  deno- 
minator of  a  fraction  are  both  multiplied  or  divided  by 
the  same  number,  the  value  of  the  fraction  remains 
unchanged. 

This  is  an  evident  consequence  of  the  combina- 
tion of  the  two  preceding  propositions,  vy^hich  show 
the  effect  of  the  multiplication  and  division  upon 
the  numerator  and  the  denominator,  to  be  exactly  op- 
posite, and  therefore,  when  performed  with  the  same 
number,  they  exactly  compensate  each  other  ;  that 
is  to  say :  as  many  times  as  the  value  of  tlie  frac- 
tion becomes  larger  or  smaller^  by  the  multiplication 
or  division  of  the  numerator  of  the  fraction,  so  many 
times  it  becomes  again  smaller  or  larger,  by  the  mul- 
tiplication or  division  of  the  denominators* 

7X9        7         7:9 

Example.      =  —  =s 

18X9        18       18:9 
where   the  mutual  destruction  of  the  effect,  of  the 
two  operations,  is  self-evident. 

The  two  first  propositions  solve  directly  all 
multiplication  or  division  of  fractions  by  whole 
numbers,  in  a  double  manner  ;  for  we  have,  evident- 
ly, every  time,  the  choice  between  two  operations, 
each  of  which  may,  according  to  the  case,  present 
a  preference  in  application. 

The  third  proposition  will  evidently  furnish  us 
the  means  to  reduce  fractions  from  one  denomina- 
tor to  certain  other  ones,  in  order  to  obtain  the 
fractional  parts  expressed  so  as  to  be  adapted  to  ccr- 


I 


VULGAR   FRACTIONS.  43 

tain  purposes  in  tlie  operations  of  arithmetic,  with- 
out changing  their  value. 

§  39.  The  investigations  of  §  37,  have  shown 
fractions  to  be  eciuivalent  to  the  product  of  a  whole 
number  into  certain  quantities  expressed  in  parts 
of  the  unit ;  when  thus  representing  quantities  of 
different  values  or  kinds,  they  have  different  denomi- 
nators ;  their  numerators  therefore  cannot  be  taken 
into  one  sum,  or  difference,  without  previous  appro- 
priate changes.  By  the  tlurd  of  the  foregoing  pro- 
positions, we  have  obtained  means  to  make  such 
changes,  without  altering  the  value  of  the  fractions. 
The  aim  of  such  a  change,  must  evidently  be  to  ob- 
tain the  same  denomination  for  both,  or  all  the 
fractions,  whose  sum  or  difference  is  desired. 

We  have  seen  in  multiplication,  that  it  is  in- 
different which  of  the  two  factors  is  multiplier  or 
multiplicand,  this  shows  that  equal  denominators 
may  he  obtained  for  two  fractions,  by  multiplying 
the  denominators  together  ;  if  therefore,  the  numera- 
tors of  the  two  fractions  are  also  multiplied,  each  al- 
ternately by  the  denominator  of  the  other,  the  value 
of  the  fraction  will  remain  unchanged,  according 
to  the  third  proposition  above  ;  and  if  more  frac- 
tions are  concerned,  considering  the  first  result  as 
one,  and  operating  upon  it  in  conjunction  with  ano- 
ther, exactly  in  the  same  way  as  before,  and  so  on 
to  the  end,  a  result  is  evidently  obtained,  that  ap- 
plies to  any  number  of  fractions.  This  furnishes 
us  with  the  following  general  rule. 

To  reduce  fractions  to  a  common  denominator  ;  mul- 
tiply the  numerator  and  denominator  of  each  fraction 
by  all  the  denominators  except  its  own  ;  then  all  the 
fractions  will  have  the  same  denominator,  and  the 
numerators  will  be  such  that  the  value  of  the  frac- 
tions will  not  be  changed. 


44  TriGAR   TRACTIONS, 

7  3 

Example,  and reduced  to  the  same  denominator 

15  14 

7  X  14  3X15 

will  give,  by  the  above        and  — — ; 

14  X  15         14  X   16 
98  45 

or ,      and       ; 

210  210 

Being  given  to  reduce  to  the  same  denominator  ; 
12      3      7 

»   ~" »        » 
2        3      5      8 

we  evidently  obtain  step,  by  step,  the  following  results  : 

3  2X2 


2x3 

3X2 

> 

3 

4 

or,       ~ 

6 

and      —  ; 

6 

from  these  and  the  third 

3x5 

4X5 

3X6 

6  X  6   ' 
20 

6x5 
18 

from  these  and  the  last, 

or,      ~  ; 
30 

30 

30 

15  X  8      20  X  8 

18  X  8 

30  X  7 

30  X  8      30  X  8 

30  X  8 

30  X  8 

120                 IriO 

144 

210 

or,  ;         ; 

240              240 

2!i) 

2)0 

Here  quantities  of  the  same  kind,  are  evidently  ob- 
tained, say  equal  parts  of  the  unit,  only  in  differ- 
ent quantities  j  for,  according  to  what  has  been  seen 
abovey  these  fractions  might  be  thus  written  : 

1111 

120  X  ;  160  X ;  144  X  ;  210X ; 

240        240        240       240 


VULGAR   FRACTIONS.  45 

§  40.  It  is  evident  from  the  above,  that  fractions 
cannot  be  reduced  to  any  denominator  indiscrimi- 
nately, as  the  new  denominator  must  be  a  multiple 
or  a  quotient,  of  the  former  denominator. 

If  it  should  become  necessary  to  take  whole  num- 
bers under  the  same  consideration,  it  will  easily 
be  judged,  from  what  has  been  said,  that 
they  must  be  considered  as  having  the  denomina- 
nator,  1,  and  such  indeed  they  are,  for  the  unit  is 
their  measure  as  to  quantity,  like  any  other  denomi- 
nator in  a  fraction, 

34  1 

Example.      34  =  —  =  34  X  —  ; 
1  1 

For  every  whole  number  whatsoever,  must  be  considered 
as  multiplied  by  1,  really  to  be  a  quantity  :  if  it  was  mul- 
tiplied by  0,  it  vvoiild  be  said  not  to  beat  all,  asO,  denotes 
the  absence  of  all  quantitv  ;  and  if  multiplied  by  any 
other  number,  the  product  would  be  another  number. 

5  41.  The  continued  multiplication  of  all  the  de- 
nominators evidently  leads  into  large  numbers,  both 
for  the  numerators  and  tlie  denominators,  which  it 
is  desirable  to  avoid  wherever  possible ;  this  will 
be  the  case  when  some  of  the  denominators  are  pro- 
ducts of  the  same  number  with  different  numbers, 
or  have  what  is  called,  common  factors^  these  are 
therefore  not  necessary  to  be  repeated  in  the  con- 
tinued product  of  the  denominators,  which  furnishes 
the  new  denominator,  as  tlie  above  example  already 
shows,  where  2  and  8,  are  products  of  2,  the  first 
by  1,  the  second  by  4. 

The  following  problem  and  its  solution,  which 
will  best  be  explained  immediately  by  an  example, 
will  lead  to  this  result. 

Problem.  To  find  the  smallest  number  which 
will  be  divisible  by  several  other  given  numbers. 

Solution*   Write  the  numbers   after  each  other, 


46  VI716AB   FRACTIONS. 

as 


3  ;  4   ;  9   ;    iO  ;   Z\    ;   35   ;    12 

1   ,  4   .   3  ,   10  ,     7     ,  36  ,    4 

3 

1   ,   1    ,  3  ,   10  ,     7     ,  35  ,     1 

4 

1,1,3,2,7,7,1 

5 

1,^,3,2,1,1,1 

7 

take  any  one  number,  w  hich  will  divide  several  ol' 
these  numbers  without  remninder,  and  divide  these 
numbers  by  it,  write  the  divisor,  here  3,  on  the  other 
side  of  a  vertical  line,  the  quotient  under  each  ot* 
the  numbers;  write  also  all  the  other  numbers,  that 
are  not  divisible,  down  in  the  line,  (as  shown  here  in 
thesecond  line  of  figures.)  with  the  quotients,  and  the 
other  numbers  pi-oceed  as  before ;  here  we  find  the 
common  divisor  4,  and  the  third  line  of  numbers  is 
obtained,  this  line  is  reduced  by  the  divisor  5,  and 
the  fourth  row  of  figur-es  is  obtained,  and  the  opera- 
tion is  continued  in  the  same  wav,  until  no  common 
divisor  is  found;  as  in  the  fifth  line  of  the  example. 
The  continued  product,  that  is  here 

3X2x7X5X4x3  =  2520 

will  be  the  smallest  number  divisible  without  re- 
mainder by  all  the  given  numbers.  The  units  of 
course  disappear  in  the  multiplication,  as  they  do 
not  augment  the  product ;  they  indicate  the  num- 
ber of  reductions  obtaified  by  the  operation,  with- 
out which  the  continue<l  product  would  have  been 
=  9525600,  and  these  two  numbers  are  both  equally 
divisible  by  the  minibers  first  given,  because  those 
factors  that  have  disappeared,  are  only  such  as 
were  repeated  in  the  given  numbers,  hy  these  being 
different  multiples  of  them,  therefore"  the  division  of 
the  number  obtained  by  the  numbers  first  given,  will 
always  give  a  whole  number,  if  the  operation  has 
been  accurately  executed  ;  thus  are  obtained  in  the 
example  the  following  numbers  : 
2520  2520  25^:0  2520 

=  840  J =  630  ; =  280  ; =  252  ; 

3  4  9  10 


VUIGAR  FRACTIONS.  47 

2520       2620       2520 

=  120  ; =  72  ;  =  210  ; 

21         36         \9, 

§  42.  If  therefore  fractions,  having  the  denomi- 
nators above  stated,  were  tc»be  brought  under  the 
same  denominator,  the  quotients  arising  from  the 
division  of  the  new  general  denominator  by  all  the 
first  denominators  successively,  vvill  give  the  num- 
bers by  which  each  fraction  is  to  be  multiplied  in 
numerat  t  and  tlenoniiMator,  to  reduce  it  to  that  com- 
mon denominator.  The  following  fractions  would 
therefore  be  changed,  as  presented  by  the  follow- 
ing operation : 

1  H  2  7  4  8  5 


10     '     21     '     36    '     12 


840       3  X  630     2  X  280     7  X  252     4  X   120 


2520    *      2520     '      2520     '      2520      '      2620 
8  X  72     6  X  210 


2520    '       2520 
846        1890     660        1764  480       576        1050 


2520 '   2520  '  2520*  2520  '        25-2o'    2520      2520 
Thus  the  fractions  are  all  brought  to  present  equal 
parts,  of  a  (lenomination  inferior  to  the  continued 
product  of  the  original  denouunators,  and  capable  of 
being  added,  or  subtracted  like  whole  numbers. 

§43.  To  add  fractions  tgether  By  the  preceding 
sections  the  fractiojjs  iiave  heen  brought  to  a  shape 
which  admits  their  being  added  and  subtracted  like 
whole  numbers,  as  they  have  shown  how  fractions 
can  be  made  to  present  the  same  parts,  or  to  be  quan- 
tities of  the  same  kind,  without  changing  their  value  ; 
thus  the  rule  to  execute  an  addition  of  fractions,  is 
"'^w  easily  deduced,  as  follows  : 


48  VULGAR  PRACTIOXS. 

Reduce  the  fractions  to  a  common  denominator, 
add  the  resulting  new  numerators,  and  give  the  sum 
the  new  denominator. 

1     7  8         7x5      43 

1st  Example.  Add  — | —  will  result h =  — 

58  5x65x8    40 

3 

=  1  -| if  the  division  is  executed  as  it  can  be  done. 

40 

2  7         2x9     3x7     18  +  21 
2d  Examp.  To  add 1 = 1 = 

3  9        3X93X9  27 
39                  12 

27  27 

The  numerator  and  denominator  of  the  fractional  part 
both  admit  of  division,  by  3,  and  the  sum  becomes  by  it, 
=  1  -f*  A.  This  application  of  the  3d  proposition  of  §  38, 
is  to  be  made,  whenever  admissible,  at  the  end  of  any 
operation  upon  fractions  ;  because  fractions  are  always 
to  be  presented  in  their  lowest  denomination. 

3d  Example.  Suppose  the  fractions  given  to  be  added, 
upon  which  the  reduction  to  the  same  denominator  has 
been  performed  in  the  preceding  §.  The  following  will 
be  the  results  successively,  being  given 

13  2  7  4  8  5 
-+-  +  -  +  --  +  -  +  -  +  - 
3         4         9         10       21       35        12 

and  making  the  sum  of  the  new  denominators  obtained 
before,  we  have  the  following  addition  of  whole  numbers 
to  make, 

840 

1890 

560 

1764 

840 

576 

1050 

7520 


SUBTRACTION   OF  TRACTIOXS.  49 

7520 

o-ifing  the  total  fractional  sum,     which  being   an 

2520 

2480 

improper  fraction,  gives  2  H ;   the   fractional  part 

2520 

is  reducible,  as  follows,  by  40, 
40 

2480     62     .  6i.' 

as —    therefore,  the  ultimate  sum  is  =  2  +  — 

2520     63  ...  63 

§  44.  SUBTRACTION  oVfrACTIONS. 
This  differs  from  their  addition  only  in  the  second 
part,  as  may  easily  be  inferred  from  all  the  preced- 
ing reasoning ;  we  obtain  therefore  the  rule :  Re- 
duce the  fractions  to  a  common  denominator,  sub- 
tract the  new  numerators  from  each  other,  and  give  to 
the  remainder  the  new  denominator. 

The  proof  of  this  rule  is  evident ;  by  bringing  the 
fractions  to  the  same  denominators  the  operation 
is  reduced  to  the  subtraction  of  the  whole  numbers, 
expressing  the  numerators,  as  is  done  in  the  addition. 

Example.    To  subtract  as  indicated  here  : 
7         2         3X7       2X9 


9 

3 

3X9       3 

X 

9 

21 

18 

3 

1 

=r 



= 

— —      = 



27 

27 

27 

9 

?d  Example, 

7 
8 

1 
5 

7X5 
5X8 

5 

8 
X  8 

50 


SUBTRACTieX   OF  FRACTIONS. 

35         8         27 


40 


40 


40 


Which  process  is  evident  by  mere  inspection,  com- 
pared with  the  rules,  and  supported  by  all  the  preced- 
ing reasoning. 

§  45.  The  total  value  of  a  number  of  fractions, 
of  which  some  are  to  be  added,  and  others  subtract- 
ed, may  thus  be  taken  in  one,  and  under  the  small- 
est denominator,  with  this  difference  only  :  that  a 
separate  sum  is  to  be  made  of  all  the  new  numerators 
to  be  added,  and  all  those  to  be  subtracted,  and  the 
sum  of  these  latter  to  be  subtracted  from  the  former, 
for  the  new  numerator. 

The  following  example  will  show  the  process. 

113         7         6         4         5         8 

_4. +-+ + 

3         6  5        16        n        27       18       25 

being  given  ;  find  the  smallest  number  divisible  by  all  the 
denominators. 


5  , 
1  , 
1  , 
1   , 


15 

5 

1 

1 

1 

27 
9 
9 
9 
3 


18 
6 
6 
3 

1 


,25 
,  26 

,   5 
,   6 


the  new  denominator  is, 

=  11X3X5X3x2x5x3=  14850 

the  successive  multipliers  of  the  fractions  are, 

14850  14850  14850 

=  2475;   =  2970; 

5 

14850 

27 


3 

■x;7<jv; 

6 

14850 
15 

= 

990;. 

14850 
11 

=  1350; 


=550; 


STJBTRACTIOli^   OF  IPRACTIONS.  51 

14850  14850 

=  825  ; =  694  ; 

18  25 

Forming  now  the  new  numerators,  by  multiplying  the  old 
ones  by  their  respective  numbers,  just  found,  and  bring- 
ing those  that  are  to  be  added  into  one  column,  and  those 
that  are  to  be  subtracted  into  another  column,  then  taking 
their  difference  for  the  resulting  numerator,  we  obtain  : 

4-  4950  ~     8910 

2475  2200 

6930  4752 

8100  

4125  15862 


4-  26580 
-  15862 


10718 
the  fraction  resulting  is  therefore  : 

26580  -  15862        10718 


14850  14850 

This  fraction  may  be  still  further  reduced  ;  the  mode  of 
doing  this  at  once  to  the  greatest  extent,  by  finding  the 
greatest  common  divisor  of  the  numerator  and  the  de- 
nominator will  be  shown  hereafter  ;  in  the  example  the 
division  by  2  is  admissible,  and  we  obtain  by  it, 

2 


10718 


14850 


5359 


7425 


Though  we  had,  in  the  above  examples,  taken 
the  smallest  number  divisible  by  all  the  denomina- 
tors, the  ultimate  fraction  was  still  reducible  ;  this 
arises  from  the  individual  circumstance  of  the  re- 
sulting numerator  being  such  as  to  have  a  multi- 
plier common  with  the  denominator,  in  the  same 
manner  as  the  denominator,  first  given,  had. 


5^  MUJLTIPX.ICATION'  or  TRACTION  b. 

§  46.  MULTIPLICATION  OF  FRACTIONS. 

The  close  connection  of  the  subject  of  fractions,  con- 
fHdered  as  unexecuted  divisions,  with  division,  and 
consequently  with  its  opposite,  multiplication,  ren- 
ders the  operations  of  multiplication  and  division  of 
fractions  more  simple  than  their  addition  and  sub- 
traction. 

For  their  multiplication  the  rule  is  simply. 
Multiply  the  numerators  into  the  numerators,  and 
the  denominators  into  the  denominators,  the  resulting 
fraction  will  be  the  product  of  the  fractions  multiplied. 
The  proof  of  this  rule  lies  in  the  two  first  ele- 
mentary propositions  upon  fractions,  stated  in  $  38  : 
for,  hy  multiplying  a  fraction  by  the  numerator  of 
another,  this  fraction  has  been  made  as  many  times 
larger  as  the  numerator  employed  indicates;  but  as 
it  was  required  to  multiply  it  by  a  number,  as  many 
times  smaller  than  this  number,  as  the  denomina- 
tor of  the  fraction,  whose  numerator  has  been  em- 
ployed, indicates,  the  multiplication  of  the  denomina- 
tor by  the  denominator  of  that  fraction  makes 
the  value  of  the  resulting  fraction  just  as  many 
times  smaller,  as  is  required. 

7  3 
Example.  To  multiply  the  fractions  —  and  —  ;  into  each 

8  10 

7  3 

other ;    or,    to  execute  -  X  —   the    multiplication    of 

8  10 

7  3  X  7        21 

-  by  3  gives =  —  ;  multiplying  then  the   de- 

8  8  8 

21  21  21 

nominator  of—  by  10,  or  making =  —  ;  the  result 

8  8  X  10     80 

is  the  value  of  the  multiplication  desired, 
in  like  manner  the  following  result  is  obtained  : 
3        6        3X6         18        9 

8        11       8X11        88       44 


DIVISION"   OF   TRACTION  S.  O 

this  last  by  reducing  the  fraction,  by   the  division  of  the 
numerator  and  denominator  by  2. 

5  47.  DIVISION  OF  FRACTIONS.  Accord- 
ing to  the  principles  and  propositions  presented  in 
the  beginning,  division  may  evidently  be  perform- 
ed by  dividing  the  numerator  of  the  dividend  by 
the  numerator  of  the  divisor,  and  the  denominator 
of  the  dividend  by  the  denominator  of  the  divisor. 
But  as  this  operation  would  often  give  fractional 
results  for  the  new  numerator  and  denominator,  it 
is  not  employed  ;  and  the  principle :  that  division  is 
the  inverse  of  multiplication,  is  here  made  use  of,  in 
concordance  with  the  two  first  propositions  repect- 
ing  fractions,  of  §  38,  from  which  is  deduced  the 
following  rule : 

Multiply  the  numerator  of  the  dividend  hij  the  de- 
nominator  of  the  divisor,  and  the  denominator  of  the 
dividend  by  the  numerator  of  the  divisor;  the  first 
gives  the  numerator,  the  second  the  denominator  of 
the  result. 

To  prove  this,  we  need  only  invert  the  reasoning 
used  in  Multiplication  ;  by  multiplying  the  denomi- 
Jiator  of  the  dividend  by  the  numerator  of  the  di- 
visor, the  fraction  has  been  made  as  many  times  too 
small,  as  the  denominator  of  the  divisor  indicates  ; 
and  by  the  multiplication  of  the  numerator  of  the 
dividend  by  the  denominator  of  the  divisor,  the  value 
of  the  fraction  is  again  made  as  many  times  larger  ; 
so  that  the  ultimate  result  presents  the  real  value  of 
the  quotient. 

4  3  4    3 
Example,     To  divide       by  -  ;  or  to  execute  -•  :  -  ; 

5  8  5     8 
4 


we  obtain,  by  the  inverted  multiplication,  at  first 

5  X 
4 
ihree  times  larger  than  -,  and  eight  times  too  small  j 
5 
•    5* 


^ 


54  DIVISION"  OT  FHACTIONS, 

multiplying  therefore  this  resultby  8,  we  obtain, 

4  X  8         32  2 

'  =  - —  =  2  +  — .  by  division;  that  is   to   say. 

5  X  3         15  15 

3                        ^2 
the  fraction  -  is  contained  |  2  -i 1  times  in  the  frac- 


8 
4 
lion  -. 


(-f;) 


It  is  evident,  that  the  result  of  such  a  division  may 
give  a  whole  number,  as  well  as  the  division  of  whole 
numbers,  for  a  fraction  can  be  contained  a  whole  num- 
ber of  times  in  another  as  well  as  a  whole  number.  As 
for  example : 

3  1         3x8  24 

4  8  4  4 

We  may  also  proceed  by  the  principle  of  the  third 
proposition  alone  ;  namely,  the  reduction  of  a  fraction, 
without  changing  its  value.  For  that  we  must  write  th? 
intended  division  fully  out  in  the  form  of  a  fraction  i/i 
numerator  and  in  denominator  j  so  the  example  above 
would  stand, 

4 


5       8  3 

8 

It  is  evident,  that  when  we  multiply  here,  both  in  nu- 
merator and  denominator,  by  the  divisors  or  denomina- 
tors of  the  individual  fractions,  in  numerator  and  denomi- 
nator, we  shall  compensate  these  divisors  or  denomina- 
tors, and  have  the  numerators  affected,  alternately,  the 
one  by  the  multiplication  of  the  denominator  of  the 
other,  thus  : 


DIVISION  OF  TRACTIOlSrs.  55 

8X4 


5X8     8X4     32 


3X5     3x5 


6x8 


As  in  bringing  the  fractions  to  the  same  denominator, 
they  of  course  compensate  in  numerator  and  denomina- 
tor, as  shown  in  the  3d  Proposition. 

§  48.  It  is  proper  here  to  add  some  remarks  upon 
the  manner  of  proceeding  in  certain  cases,  to  faci- 
litate the  calculation  of  fractions,  as  much  useless 
and  tedious  calculation  may  be  avoided  by  some  at- 
tention to  the  relation  of  the  numbers  given,  and  the 
reductions  which  they  may  thereby  present;  tliis 
operation,  by  easing  the  calculation,  will  also  make 
it  less  liable  to  mistakes. 

First.  If  any  given  fraction  is  not  reduced  to  its 
simplest  expression,  it  is  proper  to  reduce  it  pre- 
vious to  performing  the  operations  required ;  as  for 
♦'xample  : 

3        .4         6 

It  being  given  to  make j- 1 ,  the  fractions 

12         20       18 

are  immediately  to  be  reduced  by  the  equal  divisions  of 
numerators  and  denominators,  that  are  evidently  possible, 
to  the  following : 

1  11 

-   +   -   +   -- 

4  5  3 

wliich  arc  then  to  be  added  according  to  the  principles 
ibove  given. 

Second.  In  the  multiplication  or  division  of  frac- 
tions it  may  occasionally  occur,  that  such  multi- 
jdications  or  divisions  as  would  compensate  each 


56  DIVISION  or  FRACTIONS. 

other  in  tlie  ultimate  result,  may  be  avoided  by 
some  attention,  and  that  advantage  may  be  taken  of 
the  numbers  that  may  at  once  effect  a  reduction  by 
a  division  of  the  one  term,  instead  of  a  multiplica- 
of  the  other  ^  as  for  instance  : 

3  4 

—    X     —  :  the  denominators  of  each  being  respectively 

8  9 

multiples  of  the  numerators  of  the  other,  the  multiplica- 
tion is  useless,  and  the  division  may  be  made  alternately, 
the  4  being  contained  twice  in  8,  and  the  3  thrice  in  the 
9  ;  so  that  it  can  be  written  immediately, 

1       1       1 

3        2       6 

3      6 
or  in  division  forexample  ;  given  -  :   —  which  inverted 

5     25 

3      25 
into  a  multiplication  -X—    evidently  presents,  for  the 
6       6 

1  5 

same  reason  as  before,  -  X6  =  -  =  24"    2 

2  2 

Third,  Though  we  shall  hereafter  show  how  a 
fraction  may  at  one  division  be  reduced  to  its  small- 
est expression,  by  finding  the  greatest  common  mea- 
sure, it  is  often  quicker  to  make  this  reduction  by 
partial  steps,  which  may  present  themselves  to  the 
tye  at  once  ;  as  for  instance  ;  all  even  numbers  are 
divisible  by  2,  all  lO's  and  5's,  by  5  ;  the  possibility 
of  a  division  by  3,  is  often  easily  discovered;  when 
numerator  and  denominator  end  in  a  0,  it  is  evident 
that  they  are  at  once  to  be  omitted,  by  which  a 
division  by  10  is  effected  -,  and  so  on. 


i 


BEDrCTIOJf   OF  mACTIONS.  57 

So  for  example  may  be  dene  with  the  following  fraction, 
where  the  successive  divisors  are  marked  above  the  par- 
tition line  between  the  successively  reduced  fractions. 


10 


2 


1260 

126 

294 

63 

147 

9 

21 

3 

2940 

7 

or      c 
I       1296 

432 

2 
216 

9 

24^ 

4   G 

3888 

1296 

648 

72 

18 

3 

§  49.   REDUCTION   OF  FRACTIONS.     It 

is  evident,  that  the  reduction  of  fraction  is  to  be  done 
according  to  the  principles  of  the  third  proposition 
of  §  SB  ^  and  it  is  also  evident,  as  just  stated ;  that 
the  greatest  divisor  will  also  effect  the  greatest  re- 
duction. 

In  order  that  this  reduction  shall  be  accurate,  it 
is  necessary  that  the  divisor  shall  divide  both  the 
numerator  and  the  denominator  without  a  remain- 
der ;  such  a  divisor  is  called  :  the  common  measure  of 
ilie  two  numbers. 

This  requires  the  solution  of  the  following 

Problem.  To  find  the  greatest  common  measure 
<}J  two  given  numlers. 

In  order  to  present  the  principles  of  this  opera- 
tion in  the  clearest  light,  it  is  best  to  represent  the 
two  quantities  as  two  linear  dimensions;  this 
may  evidently  be  done,  as  any  quantity  may  be  re- 
presented by  a  line  containing  as  many  units  of 
measure,  as  itself  contains  units  of  abstract  quantity. 
Let  therefore  the  two  lines  JiB,  and  CD^  rqprc- 
•  nt  the  two  numbers, 


58  KEDUCTION  OF  TEACTIONS. 

a  b  c  f 

-^i 1 1 i-T-i-s=r 


fM-r-HD  =  5 

It  is  clear  that  there  can  he  no  greater  number 
that  will  divide  the  two  numbers,  (or  no  greater  line 
dividing  the  two  lines)  without  a  remainder,  than 
the  smaller  number  (or  line)  itself;  dividing 
therefore  the  greater  number  by  the  smaller,  (or 
taking  the  smaller  line  from  the  greater,  as  often  as 
possible)  as  in  the  above  CD  =  Aa  =  ab  =6c,  or 
three  times,  the  cB  remains,  smaller  than  CD,  pre- 
sents therefore  the  remainder  not  divisible  by  CD, 
and  smaller  than  it. 

But  the  number  which  can  divide  both  •&JB  and 
CD,  without  remainder,  must  also  divide  this  re- 
mainder cB ;  it  must  at  the  same  time  also  divide 
CDf  as  in  that  case  it  would  divide  its  equal  Aa, 
ab,  and  6c;  between  these  two  cB  and  CD ;  the 
same  reasoning,  used  above,  applies  again ;  namely  : 
that  they  can  have  no  greater  common  measure 
than  the  smaller  cB,  itself;  therefore,  divide  CD, 
by  cB ;  let  us  suppose,  that  as  in  the  figure  it  is 
contained  twice  in  it,  or  cJB  =  Cd  =  dg,  and  leaves 
again  the  remainder  ^1>,  smaller  than  cB;  between 
this  remainder  and  the  former  divisor,  or  the  gD, 
and  the  cB,  the  same  reasoning  takes  place  as  be- 
fore, their  greatest  common  measure  could  only  be 
the^Z>,  itself;  dividing  therefore  the  cB,  by  the 
gD,  and  suppose  it  is  contained  exactly  twice  in  it, 
or  gD  =  cf=fB,  and  leaves  no  remainder  ;  then 
this  gD,  that  is  the  last  divisor,  will  be  the  great- 
est common  measure  possible,  of  the  two  numbers 
represented  by  AB,  and  Ci>;  (it  may  be  observed, 
that  this  operation  is  to  be  continued,  as  long  as  a 
remainder  is  obtained  by  these  successive  divisions.) 
JBecausethe^D,  measures  the  cBy  without  remain- 


HEDirCTlOJfl^   OF  FSACTIOWS.  59 

der,  and  this  cB  measures  the  Cg,  the  ^D  mea- 
sures also  the  CD  ;  th  e  Ac  being  a  multiple  of  CD, 
is  therefore  also  measured  by  gD,  and  the  other  part 
of  ABf  namely  cB,  being  also  measured  by  it, 
the  whole  AB  is  measured  by  gD,  which  measures 
also  CD;  therefore,  it  is  their  common  measure,  and 
as  we  have  always  proceeded  by  the  greatest  num- 
her,  which  possibly  could  divide  the  two  numbers 
successively  given,  it  is  also  the  greatest  common 
measure,  as  was  required. 

If  no  divisor  is  found,  except  the  unit,  the  two 
numbers  have  no  greater  common  measure  ,•  that  is, 
they  are  prime  to  each  other. 

To  apply  this  to  numbers,  let  the  fraction  given  be  the 
following : 
46 

—  ;  dividing  the  denominator  by  the  numerator,  the  las*, 
153 

being  always  the  smaller  number  in  a  proper  fraction, 
(of  which  alone  there  can  here  be  question,  because  an 
improper  fraction  must  first  be  reduced,  by  dividing  b}' 
its  denominator)  we  make  the  result  of 
163  18 

—  =  3  -| —  that  is :  we  obtain  the  first  quotient  3,  and  the 
46  45 

18 
remainder    —  ;  this  fraction  inverted  for  the  similar  di- 
45 

45  9 

rision,  and  the  division  executed  gives  —  =  2  H ;  or 

18  18 

9 
the  quotient  2,  and  the  remainder  —  ;  which  treated  as 

18 
18 
above,  gives    —  =  2,  as  last  quotient  ;  and  proves  the 

9 
last  divisor  9,  to  be  the  greatest  common  measure^  for 
evidently,  the  9  ia  18  twicej  in  46  fire  times  j  in  163/ 


60  KEDUCTION   or   FRACTIONS. 

seventeen  times  ;  or  we  obtain  for  the  operation  of  the 
greatest  reduction  of  the  fraction, 
45:    9         5 

=  — ;  which  can  be  no  farther  divided,  or  re- 

153  :  9        17 
duced. 

In  the  habitual  mode  of  writing,  this  example  would 
$tand  thus  : 

45)153(3 

18)45(2 
9)18(2 

§  50.  If  by  the  foregoing  process,  no  number  is 
found  dividing  any  one  of  the  remainders,  succes- 
sively resulting,  without  a  remainder,  except  the 
unit,  the  numbers  are  said  to  have  no  common  mea- 
sure; or  they  are  what  is  called  prime  numbers  to 
each  other 9  and  the  fraction  is  not  exactly  reducible 
into  smaller  numbers. 

It  is  however  evident,  by  the  foregoing  process  : 
that  the  successive  division  has  always  approached 
nearer  and  nearer  to  the  real  value ;  that  the  re- 
mainders have  become  successively  smaller,  and  we 
might  say,  in  respect  to  a  given  case,  always  less 
important. 

If  the  above  operation  had  been  interrupted  at 
any  one  of  the  steps,  it  is  clear,  that  the  part  neg- 
lected, would  have  been  sl  fraction  of  the  last  subdi- 
vision,  made  by  the  division  of  the  last  (juotient  by 
the  last  remainder  ;  therefore,  so  much  the  smaller, 
the  farther  this  division  has  been  carried.  Con- 
sidering this  fraction  as  only  an  unit,  having  the 
last  quotient  for  a  divisor,  and  the  preceding  quo- 
tient as  the  whole  number  of  these  quantities,  to 
which  this  fraction  belonged,  we  shall  have  these 
reduced  to  the  improper  fraction  of  that  denomina- 
Hon,  by  multiply  ing  this  number  with  the  denomi- 

.tor,  and  adding  the  numerator,  that  is  the  unit ; 


KEDUCTIOX  OF  TRACTIONS.         61 

and  tlic  same  process  continued  to  the  beginning, 
through  all  the  quotients  obtained,  adding  always 
the  numerators  last  obtained,  will  ultimately  give 
the  numerator  and  the  denominator  of  a  fraction, 
approaching  to  the  fraction  which  is  intended  to  be 
approximated,  as  near  as  the  divisions  executed  w  ill 
admit ;  that  is  with  the  neglect  of  only  that  fraction 
of  the  last  subdivision  which  has  been  neglected,  as 
ahove  stated ;  for  if  the  division  liad  been  carried, 
fully  to  the  end,  we  would  evidently  have  obtained 
tlie  full  value  of  the  fraction,  as  shown  above.  And, 
as  it  appears  by  the  order  in  which  the  division  has 
been  made,  namely,  the  inversion  of  the  numerator 
into  a  divisor,  and  the  denominator  into  a  dividend, 
the  last  number  resulting  from  this  continued  mul- 
tiplication will  be  the  denominator  of  the  approxi- 
mate fraction,  and  that  Immediately  preceding  will 
be  the  numerator.  This  operation  may  be  expressed 
by  the  following  rule  : 

Divide  the  denominator  by  the  numerator,  and  the 
last  divisor  by  the  remainder  _,•  always  marking  the 
quotient,  as  far  as  the  approximation  is  intended  to 
be  carried  ;  (as  in  finding  the  greatest  common  mea- 
sure ;)  then  from  the  place  where  the  operation  is  in- 
terrupted, make  the  continued  product  of  all  the  quo- 
tients, adding  unity  to  thefrst  product,  and  afterwards 
always  the  last  previous  result,  until  the  first  quo- 
tient is  armed  at.  The  last  number  resulting  will 
be  the  denominator  of  the  approximate  fraction,  and 
the  one  immediately  preceding  it,  the  numerator. 

Example.     Let  the  fraction  3^,^%  ^^  approximated  : 
the  successive  division  will  give, 
98216\367459(3 

f   72814)98215(1 

25401)72814(2 

22012)25401(1 

3389)22012(6 

1678)3389(2 

33)1678(50 

The  success! i'e  approximations  will  be : 
6 


6£  KEDUCTION  OF  FRACTIONS. 

1st  approximation.  =  3  :  1,  or  the  fraction  | 

^successive  quotients,  3,  1,  2  3 

2d  „      <  continued  products,  1 1,  3,  2, 1         or    — 

(  n 

f  successive  quotients,  3,  1,  2,  1  4 

3d  "     <  continued  products,  16, 4, 3, 1, 1,  or  -^ 

I  15 

i                             3,    1,    2,1,6  27 

4th"    {  101,27,20,7,6,1,  or 

(  101 

(                             3,    1,    2,    1,    6,2                 58 
5th''    {  217,58,43,15,13,2,1     or 

(  217 

and  so  on  for  any  subsequent  approximation. 

If  the  above  division  had  been  carried  on  to  the  last 
divisor,  or  unity,  as  the  numbers  are  prime  to  each 
other,  the  original  fraction  would  have  been  obtained 
again,  thus  : 

3     I     1     j     2     1     1     I     6     |2|50  11|5ll|l|2j 
367459|98215|72814j2540l|22012i3389|l678|33  |  28  |  5  j  3  |  2  |  1 

the  upper  line  being  the  successive  quotients,  and  the 
lower  the  continued  products,  with  the  addition  of  the 
preceding  number*,  (or  last  numerator.) 

Suppose  the  following  fraction,  (to  give  one  more  ex 
ample,)  which  represents  the  numbers  expressing  the 
diameter  and  the  circumference  of  a  circle, 
100000000 


314169265 
The  division  gives  as  follows  : 

lOOOOOOOO'jS  14159265(3 

^  14159265)100000000(7  I 

885145)14159265(15 
5307815 
882090)885l45(t 
3055)882090(288 
27109 
26690 
2250)3055(1 

805)2250(2 

640)805(1 

165     &c.  ^ 


DECIMAL  FRACTIONS < 


65 


Ut  approximaiion.  3,1      or  - 

3 


2d 


3d 


4th  Approximation. 

i    3    ,    7    ,15,1 
)  355,  113,  16,  1,1 


J  3  ,7 
i  22  ,  7,  1 

i    3    ,    7     ,15 

i  333,  106  ,  15, 


22 


1   or 


lOG 


333 


113 


or 


355 


5th  Approximation, 
^       3       ,      7      ,    15    ,    1     ,288 


32650 


102573  ,  32650  ,  4623  ,  289  ,  288  , 1  or  1Q2573 


&c. 


CHAPTER  V. 


^K*  Of  Decimal  Fractions, 

§  51.  In  explaining  the  decimal  system  of  oiir 
usual  arithmetic,  we  have  seen  ;  that  every  figure 
designates  a  quantity  ten  times  greater,  when  it 
stands  one  place  farther  to  the  left  hand  from  the 
unit,  than  when  in  the  place  preceding  it,,  and 
therefore  conversely,  the  figure  in  the  next  place 

I      to  the   right  hand  is  ten   times   smaller  than  the 
same  figure  in  the  next  place  to  the  left  of  it. 
If  we  continue  this  reasoning  helow  the  unit  of 

j  whole  numhers,  and  after  having  marked  that  place 
by  a  (,)  and  give  denominations  to  the  parts  of  unit, 
according  to  the  same  system,  we  shall  get  succes- 
sively for  the  resulting  places,  the  denominations  of 
tenth  parts,  (or  j\),  hundredth  parts,  (or,^o^),  thou- 
sandth parts,  (ory^Vo)»  and  so  on,  to  any  part  or 
subdivision,  however  minute,  of  the  decimal  sys- 
tem J  as  for  instance,  3,45672  would  be  3  units^ 


64  DECIMAI.  FRACTIONSi 

4  tenths,    (or  VV),  5  hundredths,    (t|o)»     ^  thou- 

andths,    (loVo)?    7^   ten    thousandths,    2    hundred 

thousandths,  or  the  whole  would  be, 

4           5             6                7                  2 
3  _1-   __  J j j L_ 

10  100  1000  10,000  100,000 
where  it  is  evident  that  the  writing  of  the  denomi- 
nators can  be  spared,  because  the  successive  di- 
minution of  value  of  the  places  is  known  by  the 
system ;  therefore  the  usual,  and  easiest,  way  of 
reading  these  fractions  is,  after  having  mentioned 
the  wiiole  numbers,  to  mention  the  (,)  and  read  the 
subsequent  numbers  simply  as  tbey  occur,  leaving 
the  denominations  out,  as  understood  from  the  prin- 
ciples of  the  system.  We  have  hence  an  easy  mode 
of  expressing  any  fraction  in  the  same  system  as 
the  whole  numbers,  either  in  full  or  by  approxima- 
tions to  any  desiralile  extent. 

§  52,  AVe  have  already  seen  that  proper  vulgar 
fractions  result  from  a  remainder  oi  a  division 
smaller  than  the  divisor,  as  a  mere  expression  of 
the  unexecuted  or  unexecutable  division  ;  by  de- 
cimal fractions  we  are,  on  the  contrary,  enabled  to 
express  these  quantities,  by  continuing  the  division, 
according  to  the  same  law  that  is  used  in  the  sys- 
tem of  numbers  ;  and  all  the  difference  between  op- 
erations with  these  quantities,  and  those  with  whole 
juimbers,  will  evidently  consist  in  pointing  out 
the  place  where  the  whole  numbers  end,  and 
these  fractions  begin ;  all  the  rules  which  will  be 
found  hereafter,  for  the  mechanical  execution  of 
the  four  rules  of  arithmetic  in  decimal  fractions, 
will  therefore  merely  relate  to  the  determination  of 
the  place  of  the  decimal  mark. 

To  continue  the  division  that  is  required  to  ob- 
tain the  decimal  fraction,  after  the  number  in  the 
dividend  has  become  smaller  than  the  divisor,  it 
becomes  necessary,  to  reduce  the  remainder  to  the 
same  kind  of  unit  which  will  follow  in  the  quotient. 


DECIMAIi  FRACTIONS.  65 

and  as  this  will  be  ten  times  smaller^  the  dividend 
^vill  represent  in  it  a  ten  times  larger  number  ;  to 
do  this  we  have  only  to  multiply  it  by  ten,  which 
is  done  by  the  simple  addition  of  a  (0)  on  its  right 
hand  side,  which  will  enable  us  to  continue  the  di- 
sion ;  as  this  takes  place  at  every  step,  it  is  only 
required  to  repeat  it  also  at  every  step,  as  in  the 
following  example : 
Let  the  division  be  continued, 
34721 

=  46,5058964,  &c. 

763 
4201 
3860 
4500 
6850 
7360 
4930 
3520 
468,  &c. 
Here,  after  having  obtained  45  as  a  whole  number, 
instead  of  expressing  the  fraction  ^||  as  a  vulgar  fraction, 
an  0  has  been    added  to  the  remainder  386,  which  pre- 
sents 3860,  and  is  again  divisible  by   the  divisor  763  ;  at 
this  place  therefore,  or  after  45  in  the  quotient,  the  dis- 
tinctive mark  or  (,)    was  placed,  and  the   division  con- 
tinued, as  in  common  numbers,   with  the  constant  addi- 
tion of  a  0,  to  every  remainder,  to  make  the  continued 
division  possible ;  at  the  remainder  45,   the  addition  of 
one  0,  making  a  dividend  still  smallerjthan  the  divisor,  the 
quotient  was  0,  as  in  any  other  division,  and  the  addition 
of  a  second  0,   making  4500   in  the   dividend,  gave  the 
quotient  6,  proceeding  in  all  such  cases  as  has  been  shown 
in  common  division  ;  the  division,  which  here   stops  at 
the  seventh  place  of  decimals,  might  perhaps  continue  in 
this  case  without  ever  closing. 

§  53.  By  the  same  principle  we  can  express  any 
vulgar  fraction  in  fractions,  either  terminated,  or 
continued  as  far  as  desired,  or  solve  the 
FiioBLEM.     To  reduce  vulgar  fractions  into  ded- 
6  # 


tit)  DECIMAL  TRACTIONS. 

dmal  fractions.    Put  an  0,  in  the  place  of  Ithe  unit 
in  the  quotient^  with  a  (,)  after  it,  and  an  0,  at  the 
right  hand  side  of  the  numerator,  divide  as  iti  com- 
mon division f    adding  to  every  remainder  an  0,  and 
C07itinui7ig  this  division  as  far  as  desired,  or  until  re- 
curring numbers  occur. 
3,0 
Example.        —  =0,42857142 
7 

20 
60 
40 
50 
10 
30 
20 
6 
Placing  the  0  in  the  quotient,  and  0  in   the   numerator^ 
the  division  is  continued    as  in  common   numbers,  with 
the  constant  addition  of  a  (0)  to  the  remainder,  until  we 
again  meet  the  same  quotient,  42,  and  remainders  2  and 
6,  which  had  been  obtained  at  first,  which  are  called  re- 
curring numbers,    and   indicates  that  the  same  series  of 
numbers  would  repeat  themselves  ;  which  may  therefore 
be  done,  as  fat  as  required,  without  calculation.  Such  se- 
ries of  recurring  numbers  are  called  circulating  decimals. 

1 
As  an  other  example  :  reduce   -  =  0,3333 

3 
10 
10 
Here  evidently  we  constantly  obtain  from  the  very  begin- 
ning, the  same  quotients  and  remainders  ;  the  calculation 
ne^ed  therefore,  not  be   continued.     Of  this  kind  are   a 
number  of  other  fractions,  called  repeating  decimals. 

In  the  following  example,  we  obtain  a  complete  expres- 
sion in  decimals, 
10 

—  =0,125  ;  which  terminates  the  division  of  itself. 
8 

20 
40 


1 


DBCIMAI  FRACTIOXS.  67 

§  54.     ADDITION  OF   DECIMAL  FRAC^ 

TIONS.*— P^acc  the  numbers  under  each  other,  in 
such  a  manner  that  the  units  may  stand  under  the 
units,  and  all  the  numbers,  at  equal  distances,  to  the 
right,  or  to  the  left  of  the  units,  may  fall  under  each 
oilier ;  then  add  them  as  in  common  numbers,  begin- 
ning at  the  right  hand  figure,  and  place  the  decimal 
mark  in  the  result,  under  the  decimal  mark  of  the 
given  numbers. 

This  rule  is  evident,  from  the  ahove  considera- 
tions ;  for  the  numbers  heing  all  of  the  same  system, 
the  carrying  of  the  individual  sum  will  be  the  same 
as  in  whole  numbers,  and  each  kind  of  number  will 
thereby  stand  in  its  proper  place ;  therefore,  also, 
the  decimal  mark  will  not  change  its  position.  If 
any  one  of  the  given  numbers  should  have  no  whole 
numbers,  in  order  to  avoid  all  ambiguity,  the  place 
of  the  unit  must  be  filled  with  a  0. 

Example.     To  make  the  sum,  or  execute  : 
3,4612  +  2,134891 

3,4612 
21,34891 


^4,81011 

Write  them  as  stated  above  ;  add  as  in  common  numbers  ; 
place  the  decimal  in  the  sum  under  the  decimal  of  the 
parts. 

Example  of  several  numbers  ;         13,76094 

0,3809673 
142,012 
0,39052 


156,6444273 


§  55.    SUBTRACTION  OF  DECIMALS 

Place  the  numbers  under  each  other,  as  in  Addition, 
(placing  the  smaller  below,)  and  subtract  as  in  com- 


68  DECIMAL  FRACTIONS. 

mon  suhtradmi,  beginning  at  the  right  hand  figure^ 
and  place  the  decimal  mark  in  the  difference  under  the 
decimal  mark  of  the  two  given  numbers. 

This  rule  is  evident  from  the  simple  considera- 
tion :  that  the  subtraction  being  the  same  as  that 
of  whole  numbers,  equal  kinds  subtract  from  equals, 
the  borrowing  goes  according  to  the  same  princi- 
ples, and  the  place  of  the  decimal  mark  does  not 
undergo  any  change. 

Examfle,     3,490864-2,74962,  write  thus  : 
3,490864 
2,74962 


0,741244 


3.490864 


There  being  no  number  to  subtract  from  the  first  4,  it 
is  unchanged  in  the  remainder,  and  the  other  numbers 
follow,  as  in  common  subtraction. 

The  proof  of  this  subtraction  is  evidently  per- 
formed in  the  same  way  as  in  whole  numbers,  by 
adding  the  remainder  and  the  subtracted  number, 
which  should  give  for  their  sum  the  number  from 
which  the  subtraction  has  been  made,  as  in  the  ex- 
ample. 

If  the  number  to  be  subtracted  has  more  deci- 
mals than  the  number  from  which  it  is  to  be  sub- 
tracted, it  is  evident :  that  the  vacant  places  must  be 
considered  as  having  each  an  0,  by  which  the  sub- 
traction of  the  lower  number  is  made  by  constant 
borrowing  towards  the  left,  until  we  reach  to  the 
first  significant  decimal. 

0,9832 

0,4986735 


9,4846265 
0,9832000 


^P  DECIMAI  rEACTIONS.  69 

Here  for  the  subtraction  of  the  first  5,  the  supposed 
lending  from  before  has  given  10,  and  the  remainder  be- 
came 5  ;  then  the  next  lending  being  again  from  before, 
but  one  borrowed  from  the  10,  leaving  only  9,  the  3  is 
subtracted  from  9,  remainder  6  ;  in  like  manner  in  the 
next  we  obtained  9 — 7  =  2;  and  then  the  first  number 
.  above,  namely  the  2,  appeared  as  diminished  by  1,  and 
(he  next  subtraction  is  made  from  11,  by  borrowing  again 
as  in  whole  numbers  :  the  rest  is  evident 

§  56.     MULTIPLICATION  OF  DECIMAL 

FRACTIONS.  From  the  principles  of  Decimal 
Fractions  it  is  evident :  that  their  multiplication  can, 
in  itself,  have  no  other  rule  than  that  of  whole  num- 
bers. In  respect  to  the  value  of  the  resulting  deci- 
mals it  is  easy  to  observe,  that  as  they  are  fractions 
with  the  denominator  10,  each  of  its  preceding 
number,  they  keep  this  quality  in  the  result,  and 
therefore  present  always  in  their  product  the  pro- 
duct of  these  quotients,  which,  though  not  express- 
ed as  a  denominator,  is  indicated  by  the  place  as- 
signed to  the  decimal  mark.  This  principle  evi- 
dently causes  the  decimal  mark  to  recede  one  place 
for  every  time  that  it  is  multiplied  by  a  decimal, 
and  in  the  multiplication  of  decimals,  distant  from 
each  other,  as  many  places  as  both  decimals  to- 
gether indicate ;  because,  the  product  of  any  tens, 
hundreds,  thousands,  &c.,  into  any  other  tens,  hun- 
dreds, thousands,  &c.,  will  be  the  unii  followed  by 
as  many  O's  as  are  contained  in  the  supposed  denomi- 
nators of  the  two  fractions  together. 

This  proves  therefore,  the  following  rule  for 
multiplying  decimal  fractions. 

Multiply  the  factors  together,  as  in  whole  numherSf 
(did  cut  off  as  many  places  of  decimals,  from  the 
right  hand  side  towards  the  left,  as  there  are  decimals 
ill  both  the  factors,  and  place  there  the  decimal  mark. 

To  execute  this  multiplication,  it  is  best  to  begin 
by  the  unit,  and  multiply  by  the    other  numbers 


70 


DECIMAL  TRACTIONS. 


to  the  left,  and  then  to  the  right,  advancing  the  re- 
sult in  the  first  case  to  the  left,  and  receding  with 
the  results  of  the  decimals  towards  the  right  side, 
in  the  second  case,  in  that  order  in  which  the  mul- 
tiplication with  whole  numhers  in  the  same  ranks 
would  indicate ;  the  first  result,  that  is  the  result  of 
the  unit,  will  by  this  process,  at  once  determine  the 
place  of  the  decimal. 

Example.     Multiply  3,476  x  6,82  ;  thus  : 

3,476 
6,82 

17,380 
2, 7808 
7962 

20,  24032 

Here  the  multiplication  by  5,  as  units,  has  given  to  every 
number  in  the  multiplication  the  same  place  as  it  occu- 
pied before  ;  the  following  numbers  having  receded  to- 
wards the  right,  according  to  the  rank  they  naturally 
have  already,  occupied  their  proper  place,  and  we 
have  obtained  in  the  result  6  places  of  decimals,  exactly 
as  many  as  both  factors  have  together,  and  as  the  rule 
above  prescribes  ;  (for  which  therefore  it  might  form  a 
a  practical  deduction  or  proof.) 

Let  the  following  examples  be  executed,  to  illustrate 
further  the  practical  application  and  its  consequences. 


36,  462 
0, 4937 

172,7892 
66,32 

0,  6378 
0, 0624 

14,6808 
3, 28068 
109356 
256164 

1036,  7352 
8639, 460 
61,83674 
3, 465784 

0, 032668 
10766 
21512 

0, 03355872 

17,9963524 

9731,488624 

DECIMAIi  raACTIONS.  71 

The  first  example  above  gives  a  result  that  makes  the 
product  recede  one  place  in  the  decimals,  because  the 
first  multiplier  is  a  decimal,  this  determining  the  decimal 
place,  all  the  others  follow  by  themselves,  the  results  re- 
ceding always  one  place,  and  ultimately  giving  as  many 
places  of  decimals  as  there  are  in  both  the  factors  taken 
together.  When  in  the  second  example  the  multiplica- 
tion by  6,  as  unit,  was  performed,  each  number  resulting 
kept  the  place  it  occupied  in  the  multiplicand,  the  deci- 
msd  place  being  determined  by  this  ;  the  next  multiplica- 
tion with  5,  or  properly  60,  gives  the  advance  towards 
the  left,  according  to  common  multiplication  of  whole 
numbers,  the  two  decimals  being  made  to  recede  one 
place  farther  to  the  right,  each  tends  to  keep  the  final  re- 
sult in  its  proper  order,  the  number  of  decimals  are  thus 
determined  by  the  mere  addition,  and  are  conformable 
to  the  above  rule. 

6  6 

In  the  third  example  :  the into  the  —  ;    gives,  evi- 

100-  10 

30 

dently,   :  which  assigns  to   this   result  the  place 

1000 

seen  in  the  example;  the  others  follow  by  themselves  ;  but 
if  we  had  not  attended  to  this,  the  rule  above  would  give 
the  coincidence  with  the  result  obtained,  as  may  be  easily 
seen. 

§  57,  It  is  most  generally  needless  to  calculate  to 
the^full  extent  of  all'  the  decimals  of  both  factors, 
the  one  or  the  other  factor  commonly  indicates  the 
number  of  decimals  to  which  it  is  intended  to  carry 
the  accuracy,  and  we  may  dispense  with  the  smal- 
ler decimals,  so  much  the  rather,  as  they  will  at 
all  events  not  be  absolutely  accurate,  if  the  decimal 
fractions  are  not  themselves  exact,  in  consequence 
of  the  absence  of  the  smaller  decimals,  that  would 
influence  the  places  taken  beyond  the  lowest  that  is 
given  in  one  or  the  other  factor,  for  this  reason  it 
is  desirable  to  hav£  an  easy,  and  exact  way  of  ex- 


75  DECIMAI.   FRACTIOTV-S- 

editing  this  abridged  multiplication,  which  is  as 
follows  : 

Multiply  by  the  greatest  iiumher  of  the  multvplier first, 
and  determine  the  place  of  the  decimal  ;  (as  in  the  pre- 
ceding rule ;)  then  mark  this  number,  and  also  the 
lowest  decimal  of  the  multiplicand^  then  take  the  next 
lower  number  of  the  multiplier,  and  multiply  all  the 
multiplicand  by  it,  taking  from  the  product  of  the  de- 
cimal marked  off,  only  the  part  which  is  to  be  carried 
forward  to  the  next  place,  using  the  ten  nearest  to 
the  result,  write  the  product  under  the  foregoing,  so 
that  the  first  figure  to  the  right  comes  under  the  first 
figure  of  the  foregoing  product;  thus  continue  as  long 
as  there  are  figures  in  the  multiplier,  always  marking 
off  one  figure  in  the  multiplicand  for  each  factor  of 
me  multiplier,  and  making  the  addition  of  the  carry- 
ing as  before;  the  decimal  mark  will  take  its  place  ac- 
cording to  the  determination  of  the  first  number  used 
as  a  multiplier. 

Example.  The  first  of  those  given  above,  executed 
according  to  this  rule,  will  show  that  in  this  manner  re- 
gularity is  insured,  and  the  deviation  from  the  full  result 
obtained  above  will  not  extend  farther  than  the  last  place 
of  figures,  or  rather  only  the  next  after  it,  as  it  does  not 
differ  a  whole  unit  of  that  place. 

36,452, 
0,4937 


14,6808 

3,2807 

1094 

17,9964 


4 
The  first  line  in  the  product  of  —  into  the  multiplicand. 

10 


DIVISIOX  OF  DECIMAI  FRACTIOXS.  73 

In  the  second  line  we  had  to  carry  for  the  product  of 
9x2=  18,  which  being  nearer  to  20  than  to  10,  the  tens 
to  be  carried  were  two.  So  that  when  we  afterwards  made 
9  X  6  =  45,  we  added  2,  making  47,  the  7  being  placed, 
the  4  carried,  and  the  operation  continued  as  in  common 
numbers  ;  the  rest  of  the  operation  is  exactly  the  same 
with  respect  to  the  remaining  numbers  of  the  multipli- 
cand ;  for  in  the  third  line  we  had  3x4  =  12,  and  the 
3  X  52  from  before,  giving  2  to  carry,  rather  than 
only  1 ,  we  added  2,  that  is,  we  made  1 2  -|-  2  =  1 4,  then 
continued  3  x  6=18,  and  1  carried,  =  19,  and  so  on  : 
the  addition  is  as  before.  More  examples  will  occur  in 
the  practical  part. 

§  58.  DIVISION  OF  DECIMAL  FRAC- 
TIONS. From  the  manner  the  origin  of  Decimal 
Fractions  has  been  deduced,  it  has  already  been  seen : 
that  the  decimals  began  whenever  the  divisor  was 
larger  than  the  dividend,  or  which  is  the  same  thing, 
when  an  0  became  necessary  to  be  added  ;  in  other 
words,  when  it  became  necessary  to  recede  towards 
the  right  farther  than  the  quantities  of  the  dividend 
furnished  numbers  of  the  A^alue  of  the  divisor ;  as 
many  such  steps,  therefore,  as  it  may  become  ne- 
cessary to  make  until  a  figure,  actually  significant, 
can  be  obtained  in  the  quotient,  as  many  O's  will 
precede  it,  the  0  of  the  unity  place  being  counted  as 
the  first;  or  the  place  of  the  first^significant  deci- 
mal will  be  that  indicated  by  the  number  of  steps 
it  was  necessary  to  recede. 

It  will  be  most  easy  in  this  place,  and  will  furnish 
us  with  the  clearest  method  of  accounting  for  the 
necessary  steps,  to  proceed  from  the  relation  of  the 
unit  in  the  dividend  and  the  divisor,  as  a  leading 
priiiciple  for  the  determination  of  the  decimal  mark ; 
and  in  cases  where  no  whole  number  is  obtained  in 
the  quotient,  it  will  be  entirely  referable  to  common 
division  continued  to  decimals,  as  employed  by  us 
7 


74  DIVISION  OF  DECIMAL  FRACTIONS. 

in  elucidating  the  principle  of  decimal  fractions.  The 
rule  resulting  will  therefore  be  this  : 

Begin  the  division  as  in  whole  numbers ,  and  place 
thejirst  decimal  resulting ,  as  many  places  to  the  right 
of  the  decimal  mark,  as  it  has  been  necessary  to  re- 
cede from  the  first  figure  in  the  divisor,  to  obtain  in 
the  dividend  a  number  sufficiently  large  to  be  divided 
by  the  divisor. 

Example,     Divide  3,  46921 

=  0,80173587  +  &c. 

4,327 
7610 
32830 
25410 
37750 
31340 
1059  +  &c. 

Here,  it  is  evident,  that  4  not  being  contained  in  3, 
whatever  may  be  the  following  decimals,  the  divisor  can- 
not be  contained  in  the  dividend,  and  there  is  no  whole 
number  in  the  quotient,  therefore  a  0  is  written  in  the 
place  of  the  unit ;  the  next  receding  in  the  dividend 
giving  a  significant  figure  ;  this  falls  in  the  first  place  of 
decimals.  And  now  the  division  being  performed,  as  in 
whole  numbers,  and  the  0  added  always  when  there  were 
no  numbers  more  to  be  taken  down  from  the  dividend, 
the  division  can  be  carried  as  far  as  may  be  desired,  for 
the  decimals  determine  themselves  without  any  further 
care.  But  it  is  evident  here  again  :  that  the  division  to  a 
greater  extent  than  is  warranted  by  the  numbers  given, 
will  not  give  the  full  accuracy,  when  the  decimals  are  not 
determined  ones,  and  only  approximations  ;  for  the  O's 
set  down  should  evidently  always  be  the  figures  which 
would  have  followed  in  the  dividend  ;  and  the  products 
to  be  subtracted,  should  be  affected  by  the  lower  deci- 
mals which  are  missing  in  the  divisor. 

The  two  Examples  that  follow,  will  show  more  of  this 
application. 


DIVISIOX  OF  DECIMAL  FRACTIONS.  70 

4,27G9  0,00042769 

=  0,007537+&c. =  0,007537+ 

.j67,432  0,0567432                        [&c. 

3  048760  3048760 

2116000  2116000 

4137040  *           4137040 

165016  165016 

These  ex.imples,  showing  both  cases  of  whole  num- 
bers and  decimals,  with  a  denominator  exceecling  the 
numerator,  are  both  equ.il  applications  of  the  rule 
found  above  :  and  their  result  is  th^^  same,  as  the  one  is 
evidently  the  product  of* the  other,  by  a  multiple  of  10  ; 
in  both  cases  it  was  necessary  to  recede  three  steps,  to 
obtain  a  signitir;mt  numlter  in  the  quotient;  therefore, 
the  first  numher  in  the  quotient  is  rn  the  third  place  of 
decimals  ;  or,  we  might  say,  it  was  three  times  impossi- 
ble to  divide  ;  therefore  we  have  three  O's,  the  Oof  the 
unit  place  being  counted,  as  of  course,  because  the  first 
time  the  division  was  not  posj-ible,  was  that  of  whole  units. 

In  the  same  way  as  we  have  found  in  vulgar  fractions  : 
that  one  fraction  may  be  contained  in  another  fraction  a 
whole  number  of  times,  as  well  as  one  whole  number  in 
another  ;  so,  of  course,  this  also  takes  place  in  decimal 
fractions,  as  in  the  two  followinir  examples,  which  again 
present  the  same  quotient,  as  the  two  last. 

452,9673a  .  0,045296738 

=29,580770  + &c. =  29,580770  +  Szc. 

15,3129  0,0U1531J9 
146  7093  1467093 

V,  89328  889328 

1  236U30  1236S30 

1179!'.0  n79U0 

1078930  1078930 

70270  +  &:c.  70270  +  fcc. 

These  examples  first  admitted  of  division  by  whole 
numbers,  because  15  is  evidently  contained  in  452  a 
whole  number  of  times,  which  we  found  to  be  29  :  then 
the  decimals  began  ;  these  two  whole  numbers  are  evi- 
dently easily  determined,  as  the  divisor  has  only  two  pla- 
ces of  figures,  and  the  dividend  3  in  the  corresponding 
places  of  both  examples  ;  and  these  are  already  twice 
divisible,  before  recourse  is  had  to  the  decimals  from  the 


76  DEXOMINATE    FRACTIOlSrS. 

Jividend,  for  the  subtraction  of  the  products  of  the  whole 
numbers  of  the  divisor  into  the  quotient. 

Remark.  If  it  is  required  to  perform  any  one  of  the 
four  rules  o^' arithmetic,  between  decimal  and  vulgar  frac- 
tion?, the  decimal  fractions  are  to  be  considered  as  whole 
numbers,  paying,  of  course,  due  attention  to  the  place  of 
the  decimal  mark  ;  this  will  therefore  need  no  special 
explanation.  But  it  will,  in  many  cases,  be  most  conve- 
nient to  reduce  the  vulgar  fraction  into  a  decimal  frac- 
tion, and  then  proceed  upon  the  principles  of  decimal 
fractions.  This  being  therefore  a  subject  depending  on 
the  judgment  of  the  calculator,  the  principles  of  which 
have  been  explained  sufficiently,  it  will  not  need  here  to 
be  treated  separately. 


CHAPTER  V. 

Of  Denominate  Fractions, 

§  59.  I  take  the  liberty  of  calling  all  those  subdi- 
visions of  an  accepted  unit  that  have  received  par- 
ticular names,  and  which  properly  form  fractions 
of  this  unit,  with  a  certain  conventional  denomina- 
tor, that  is  therefore  always  understood.  Deno- 
minate Fractions  ;  such  are  all  the  subdivisions  of 
measures  of  any  kind;  of  length,  surface,  or  solid- 
ity, weights,  money,  time,  &c. 

In  order  to  make  use  of  these  fractions  in  arith- 
metic, it  is  necessary  to  know  their  conventional 
denominators,  or  to  be  able  to  say  how  many  units 
of  each  subdivision  make  a  whole,  or  a  unit  of  a 
higher  subdivision;  as  for  instance,  the  general  di- 
vision of  pound  (money)  into  shillings,  pence,  and 
farthings ;  where  the  general  habit  is  that  1  shil- 
ling =  ^\  of  a  pound  ;  1  denier  or  penny  =  /^  of  a 
shilling;  1  farthing  =  ^  of  a  penny.  Or,  in  the 
other  manner  of  expressing  it, 

£1  :=  20s;  Is  =  12rf;  Id  =  4/. 


DENOMINATE   FRACTIONS.  77 

Of  these  subdivisions,  old  habits,  unconnected  in 
their  origin,  and  therefore  devoid  of  system,  have 
introduced  such  a  variety,  that  it  is  necessary  to 
have  tables  in  order  to  recall  them  to  memory; 
such  tables  will  be  placed  at  the  end  of  this  book, 
to  which  1  shall  add  tlic  approximate  or  full  decimal 
expression  of  the  unit  of  each  subdivision  in  the 
other,  and  in  the  wliolc,  as  it  is  evidently  possible 
to  express  them  all  in  decimal  fractions,  either  ex- 
actly or  approximately.  Certain  signs  have  been 
given  to  all  these  subdivisions,  to  abridge  their 
notation ;  these  will  be  learnt  from  the  tables. 

In  thus  stepping  aside  from  the  simple  theory  of 
a  system  to  a  mere  practical  habit,  we  shall  soon 
feel  what  an  advantage  it  would  be  in  all  transac- 
tions where  quantity  is  concerned,  to  have  a  regu- 
lar and  unique  system  for  them  all ;  but  the  attempt, 
so  often  made,  has  always  been  frustrated,  by  the 
unwillingness  of  men  engaged  only  in  their  private 
concerns,  to  all  mental  motion  or  exercise,  not  di- 
rectly advancing  their  private  aims.  Similar  sys- 
tems had  been  in  use  in  common  arithmetic,  before 
the  adoption  of  the  decimal  system  of  numeration, 
the  advantages  of  which  soon  expelled  them  from 
theoretic  arithmetic. 

It  is  evident  that  the  difficulties  to  be  vanquished 
in  this  part  of  arithmetic,  consist  only  in  the  atten- 
tion that  is  required  to  be  paid  to  the  effect  of  the 
irregular  system  of  subdivision,  which  determines 
the  principles  of  what  may  be  called  carrying  from 
one  denomination  to  the  other ;  the  rules  discovered 
hereafter,  therefore,  chiefly  refer  to  this  operation  ; 
they  will  not  need  any  proof,  as  they  have  only  the 
arbitrary  subdivisions  for  their  principle,  and  for* 
their  aim^  to  facilitate  the  several  processes,  '^i'hey 
will  therefore  be  given  simply,  witli  a  few  exani- 
jdes  for  illustration,  their  proof,  as  far  as  arithme- 
tic principles  are  concerned,  lying  always  in  the 
7  # 


TS  DENOMINATE    FRACTIONS. 

principles  of  calculation  already  explained;  and 
their  combination  will  be  reserved  for  the  practical 
part  of  this  treatise. 

§G0.  ADDITION  OF  DENOMINATE  FRAC- 
TIONS. Rule.  Jf^riie  the  numbers  of  each  deno- 
mination nnder  each  other,  distinguishing  them  by 
points  ;  add  them  as  whole  numbers,  beginning  at  the 
most  nght  hand  figures,  and  carry  from  one  deno- 
mination to  the  other,  according  to  the  value  of  the 
subdivision,  in  parts  of  the  next  superior  quantity, 
that  is,  by  dividing  the  sum  obtained  by  the  deno- 
minator of  the  fraction,  indicated  by  the  subdivision. 
Example — in  feet,  inches,  and  tenths  of  inches. 
12  in.  =  I  f.  affords  the  principle  of  carrying  from  inches 
into  feet ;  the  mode  oi  carrying  the  tenths  of  inches 
being  as  explained  in  decimal  arithmetic. 

To  add   12  f.  7,6in. -fSf.  4,9  in.  -f  2  f.  11,2  in. 
f.      in. 
12.    7,6 
3.    4,9 
2.  11,2 

18.  11,7 

Here  tlie  sum  of  inches  being  23,  12  are  taken  away 
to  carry  as  one  unit  to  the  feet ;  there  remain  1 1,7  inches; 
the  rest  is  exactly  like  the  addition  of  whole  numbers. 

Example  in  weight,  of  pounds  Troy  ;  the  subdivisions 
of  which  are, 

1  lb.  =  12  oz  ;   1  oz.  =  20  dwt ;   1  dwt.  =  24  gr. 

lb.  oz.  dwt.gr. 

To  add  7.  10.  14.  12 

19.    6.  17.  14 

6.  11.  15.  19 


36.    5.    7.21 

Adding  the  first  column  to  the  right,  or  of  grains,  what 
is  over  24  gives  21  to  set  under  this  denomina- 
tion, and  1  dwt.  is  carried  to  the  next  denomination,  or 


DENOMINATE    FRACTIONS.  79 

Jvvt.  column.  This  second  culumn  being  added,  gives 
47  dvvt.  =  2  oz.  +  7  dwt ;  therefore  the  7  dwt.  are 
placed  under  that  column,  and  2  oz  are  carried  to  the 
ounces  ;  the  ounces  added,  with  the  carrying,  give 
29  oz.  =  2  lb  -|-  o  oz  ;  the  latter  placed  under  ounces 
and  the  2  lb.  carried  give  ultimatel}',  by  the  addition  ot' 
the  last  column,  the  number  of  whole  units  of  pounds  3C; 
then  the  whole  sum  is  txpressed.  These  examples 
may  suffice  for  the  present,  as  more  will  appear  in  the 
practical  part. 

§  bl.  SUB  IRACTION  OF  DENOMINATE 
FRACTIONS.  Rule.  JFriie  the  denominations 
of  the  subdivisions  of  the  quantity  to  be  subtracted 
under  the  same  denominations  in  the  quantity  whence 
they  are  to  be  subtracted^  and  siibtract  in  each  column 
the  lower  from  the  upper ^  beginning  at  the  lowest 
denomination;  and  in  borrowing  from  a  superior 
denomination^  give  to  the  unit  borrowed  the  value  it 
has  in  the  lower  suhdivision  in  which  it  is  used. 

This  rule  is  evident  froin  the  simple  inversion  of 
what  is  directed  to  be  done  in  addition,  and  is  ana- 
logous to  the  rule  in  the  decimal  system,  as  is  evi- 
dent. These  subtractions  evidently  admit  of  proof, 
by  the  addition  of  the  remainder  to  the  number 
subtracted,  as  is  the  case  in  all  subtractions. 

Example  in  feet,  inches,  and  tenths, 
f.     in. 


17. 

7,3 

8. 

9,6 

8.1 

9,7 

17. 

7,3 

Example  in 

pounds  T^03^ 

lb.    oz. 

.  dwt 

•g'"' 

22.    7. 

6. 

6 

14.    9. 

16. 

12 

7.    9. 

9. 

17 

7.    6. 


80  DENOMINATE    FRACTIONS. 

The  borrowing  in  the  hrsi  example,  from  the  feet  to 
the  inches,  h.is  given  12  -f  6  in.  =  18  in  from  which  9 
taken  left  9,  the  decimals  having  been  IreHted  as  usual, 
then  16  —  8  ft.  =  8  ft.  gave  the  remainder  of  the  whole 
feet.  So  also  in  the  second  example,  ive  had  first,  by 
borrowing  to  the  Jiraiiijj,  24  +  5  ~  12  =  17  (grains)  ; 
then  in  the  second  column  20  -f-  6—  16  =  9  pennyweights, 
in  the  third  column  12+6  —  9  =  9  ounces,  and  lastly 
21  —  14  =  7  poiHKSg,  the  borrowing  having  been  made 
throughout  accordmg  to  the  dictates  of  the  arbitrary  sub- 
division, nnd  the  numbers  from  which  the  units  were 
successively  borrowed,  having  always  been  diminished 
by  a  unit 

§  62.  MULTIPLICATION  OF  DENOMI- 
NATE FRAi  'ilONS.  From  the  two  different 
ways  in  which  it  has  been  seen  that  denominate 
fractions  may  be  compared,  namely,  as  units,  of 
which  a  certain  number  form  another  unit,  or  as 
iVactions  of  the  preceding,  or  rather  the  highest 
unit,  with  a  certain  denominator  understood,  it  may 
be  inferred  :  that  the  multiplication  of  them  can  be 
executed  in  two  different  ways. 

The  first,  using  the  given  numbers  as  imits,  w  ill 
form  fractions,  of  a  denomination  adapted  to  the 
conventional  system  of  subdivision,  which  mode  is 
exactly  analogous  to  what  has  been  done  in  decimal 
fractions.     This  is  often  called  Cross  Multijdicatiun, 

The  second  consists  in  using  the  lower  units  that 
are  given  as  fractional  parts  of  the  whole,  and  takes 
the  products  of  the  multiplier  into  the  multiplicand 
as  such,  distributed  in  parts  that  are  best  adapted 
to  the  easy  division  of  the  multiplicand,  and  ex- 
presses the  results  in  the  same  unit,  and  its  subdivi- 
sions ;  the  final  result  is  obtained  by  the  addition  of 
the  products  j  this  method  is  usually  called  Practice. 

In  the  application  of  multiplication  to  this  species 
of  quantities,  we  are  limited  by  their  nature  to  those 
which  are  capable  of  producing  things  really  exist- 
ing in  nature ;  as  lineal  measures  into  each  other, 


^^whicli  produce  surfaces,  and  these  again  into  lineal 
measures,  which  produce  solids.  Or  to  such  as  arc 
of  a  different  nature  from  each  other;  and  their 
relative  possibility  of  being  compared  witii  each 
other  renilers  them  fit  to  give  a  result  existing  in 
nature,  as  for  instance,  money  into  weights,  or 
measures,  of  any  kind,  where  only  the  numbers  or 
quantities  are  usetl,  and  the  things  themselves  arc 
considered  as  capable  of  representing  each  other, 
that  is  :   conventionally  comparable. 

But  such  quantities  as  money  into  money,  weight 
into  weight,  being  incapable  of  producing  any  pos- 
sible result,  cannot  be  objects  of  this  or  any  calcu- 
lation ;  and  to  the  latter  only  the  second  method  is 
conveniently  applicable,  because  the  resulting  infe- 
rior fractions  become  of  an  irregular  computation, 
if  cross  multiplication  was  applied. 

§  63.  To  multiply  by  the  first  method,  or  Cross 
Mtdtiplicationf  we  have  the  following  rule,  grounded 
upon  the  general  principles  of  fractions. 

Multiply  all  the  columns  of  the  multiplicand  succes- 
sively by  all  the  numbers  of  the  columns  of  the  multi- 
plier, beginning  at  the  right  hand  figure^  and  carry ^ 
in  each  passage  to  a  higher  columiu  according  to  the 
value  of  the  subdivisions  made  use  of ;  place  the 
results  of  equal  quantities  under  each  other ;  their 
sum  in  the  result  will  give  the  whole  quantities  and 
the  subdivisions,  according  to  the  same  scale  as  the 
preceding  subdivisions  ;  that  is,  the  denominators  be- 
coming equally  products  cf  the  denominators  indicated 
by  the  subdivisions. 

Example,     To  multiply  7  ft.  2  in.  X  6  ft.  5  in. 
ft.  in. 
7.    2 
6.    6 


2.  11.  10 
43.    0 

46.  11.  10 


82  DENOMINATE   FRACTIONS. 

Cy  muUiphing  here  2  in.  into  5  in.  we  have  properly 

2         6  10 

made  —  x  —  =  ■ —  ;  therefore   we  have   obtained  a 
12       12        144 

subdivision  of  (he  unit  one  degree  lower  thnn  those  em- 
]>loyed,  iis  in  decinrial  fnictions  ;  therefore,  also,  in  writ- 
ing; the  rcj^ult,  this  has  been  removed  one  step  more  to 
the  rij^ht.     In  making 

5  35  11 

7  ft.  X  5  in.  =  7  X  -  ft.  =  —  ==  2  H , 

12  12  12 

^ve  have  obtained  first  twelfths,  and  then,  by  reductiou  to 

whole  numbers,  2  whole  quantities,  and  —  of  a  foot;  the 

12 

2         12 
result  of  2  in.  X  6  ft.  =  6  X  —  =  —  =  1,  hasgiren,  bj 

12  12 
the  same  principles,  one  foot  to  carry  to  the  next  result ; 
then,  by  the  uiiihiple  of  the  feet  into  the  feet,  ivith  this 
addition,  we  have  6  X  7  -j-  1  =  43  feet ;  the  final 
sum  is  obtained  as  in  addition,  but  presents  an  inferior 
subdivision  one  degree  lower  in  the  same  scale  of  subdi- 
vision that  is  used,  or  twelfths  into  twelves  ;  thence  the 

11  10 
above  result  is  =  45  -^  ■—  + • 

12  144 

If  we   multiply  again  (for  example  the  above  result) 
!)y  a  lineal  dimension,  we  shall  obtain  a  solid,  expressed 
in  the  same  system  of  subdivision  ;  thus  ; 
ft.   in.    1. 
45.11.10 
5.    7 


26.    9.  10.  10 
229.  11.    2 


25G.    9.    0.  10 
Mere  we  again  obtain,  by  the  very  same  process  as  above. 


DENOMINATE    FRACTIONS.  83 

a  denomination  still  lower  in  the  same  scale  of  subdivi- 

9  10 

sion,  or  =  256  -\ 1 . 

12       12  X  12  X  12 

§  64.  Multiplication  of  Denominate  Fractions  as 
Fractional  Parts  of  the  Whole^  or  Practice.  The 
nature  of  this  operation,  as  stated  above,  evidently 
leads  to  the  rule. 

Multiply  the  whole  and  fractional  parts  of  the  multi- 
plicand by  the  whole  numbers  of  the  multiplier  ;  then 
distribute  the  fractional  parts  of  the  multiplier  into 
such  as  are  most  easily  taken;  take  such  parts  of  the 
whole  and  fractional  part  of  the  multiplicand  as  will 
be  indicated  by  them,  and  add  all  these  parts  for  the 
final  result. 

By  this  operation  the  products  of  the  different 
fractions,  distributed  for  the  convenience  of  the  ope- 
ration, being  partially  taken,  the  proof  of  the  rule 
lies  in  the  simple  multiplication  of  fractions,  and 
this  is  only  repeated ;  it  is  generally  convenient  to 
divide  the  fractions  of  the  multiplier  so  that  the 
smaller  subsequent  parts  are  again  fractions  of  the 
first. 

For  the  purpose  of  comparison  with  the  other  mode, 
we  shall  use  the  example  already  given, 
ft.  in. 
7.    2 
6.    5 


43. 

0 

2. 

H 

=z 

f 

4 

0. 

H 

= 

I 

4 

X 

i  • 

=  1  m.  ^ 


45.111 
Here,  after  multiplying  the  7  ft.  2  in.  by  6,  the  whole 
^Timber,  giving  12  in.  -|-  42  ft.  =  43  ft.  the  5  in.  ofthe  mul- 
)Uer  are  divided  into  4  in.  =4  ft,  and  1  in.  =  i  x  4,  in. 


84  DEXOMINATE   FRACTIONS. 

or  1^  X  I  ft,  the  i  x  7  ft.  giving  2  ft.  and  1  ft.  remaining, 
which  gives  12  in.  to  add  to  the  2  in.,  the  I  of  the 
(12  +  2)  in.  =  14  in.  being  taken,  gives  4^  in.,  as  in  the 
second  line  ;  the  second  fractional  part  being  ]  of  that,  or 

2 

of  2  ft.  +  42  in.  =  28|  in.,  the  I  of  which  is  7  -\ 

3X4 
=  7  +i  in.,  gives  the  third  line  ;  the  addition  of  the  pro- 
ducts is  easily  understood  from  the  addition  effractions, 
as  I  +  I  =  f  +  i  =  f ,  and  the  rest  is  like  the  addition 
of  denominate  fractions. 

Let  this  example  be  continued,  as  was  done  in  the  for- 
mer case. 

ft.     in. 

45.1  If 

5.    7 


229.  Ill 


111  =  1=6 


S        Q7  1     — 


1  =  6  in.  }  „  . 

|x^=lin.r-- 


256.    9^'^ 

We  have  5  x  |  =  V  *=  ^  -f  i,  for  the  product  of  the 
whole  number  5  into  the  fraction  9  then  5  x  11+4  = 
59  in.  =  4  ft.  11  in.,  from  the  whole  into  the  inches, 
with  the  carrying,  the  rest  then  as  whole  numbers,  or 
45  X  6  -+-  4  feet.  For  the  7  in.  we  take  6  in.  =  i  ft,  and 
J  in.  =  i.  X  ^  ft.  ;  therefore  h  x  (45  ft.  llf  in  )  giving 
the  second  line  easily  ;  and  the  third  line  being  J-  of  that, 
presents  2_2  ft.   =  3  ft.   +  %«  in.  ;  the  second  part  or 

48  in.  +  11  in. 

. =  9  in.  +  I ;  and  the  fractional  part  of 

6  '  " 
this,  making  in  ^\  as  the  upper  fraction  is  }2,  which  add- 
11  "  71 

ed  to ,  gives =  ^. 

6  X  12  12  X  6 

The  addition  is  executed  in  the  fractional  part  by  re- 
ducing them  all  to  the  denominator  72,  which  is  their 


t 


DEN^OMINATE   FRACTIONS.  85 

)mmon  multiplier  ;  the  whole  inches  are  then  carried 
J  the  next  addition,  which  is  executed  as  shown  in 
its  place.  If  any  such  denominate  fraction  of  any  kind 
is  to  be  multiplied  by  a  whole  number,  it  is  evident, 
that  nothing  is  required  but  to  multiply  each  of  the 
parts  by  this  number  successively,  in  the  above  order  ; 
carrying  according  to  the  principle  of  the  given  arbitrary 
subdivision,  exactly  as  was  done  with  the  first,  or  whole 
number  above.  This  is  evidently  admissible  with  any 
subdivision,  and  needs  no  separate  explanation,  as  it  ha;? 
actually  been  already  given. 

§  65.  The  two  preceding  modes  of  performing  the 
multiplication  of  denominate  fractions  being  evi- 
dently cumbersome  when  applied  to  great  calcula- 
tions, and  when  the  fractional  parts,  or  lower  deno- 
minations, are  not  easy  aliquot  parts  of  the  whole, 
it  will  be  often  most  convenient,  to  reduce  the 
factors  to  whole  and  decimal  fractions,  by  the  me- 
thods taught  in  their  proper  place,  and  then  to  pro- 
ceed by  multiplication  of  decimals ;  for  this  purpose 
the  fractions  would  be  marked  with  their  divisors, 
according  to  the  habitual  subdivision. 

Example.  Let  the  preceding  question  be  executed  in- 
this  way,  and  by  abridged  multipHcation  of  decimals  ;  wo, 
obtain  as  follows  : 

7  +  ^2^  ==  7^1666  . . . .  :  6  +  /^  =  6,4166G 
Performing  the  abridged  multiplication  thus  : 

6,41666 

7,16666 

44,91666 

64167 

38500 

3850 

385 

38 

4 


45,98610 


S6  DEKTOMIN-ATE  FRACTIONS, 

As  both  fractions  are  repeating,  having  a  continued  repeti- 
tion of  the  6,  the  products  of  these  have  been  inserted, 
as  long  as  they  have  any  influence  in  the  numbers  pre- 
served, by  repeating  the  first  product  of  6,  receding 
every  time  one  place  more  to  the  right ;  and  in  the  last 
numbers  of  the  products  always  carrying  from  a  product 
of  a  previous  6  ;  this  has  been  performed  throughout, 
by  augmenting  the  last  figure  one  unit. 

The  second  multiplication,  treated  in  the  same  way, 
will,  when  executed,  give  the  following 

Example.  5  -f  ^2  =  6,58333,  the  series  of  3  be- 
ing again  continued. 

45,9861 

6,6833 

229,9305 

22,9330 

3,6789 

1379 

138 

14 

1 

256,7566  &c. 

To  compare  these  three  results  together  will  be  mofev 
easily  done  by  reducing  the  two  former  ones  to  decimals 
also ;  thus  we  obtain  by  §  60, 
9  10 

256  +  -  -} =:  256,75578  &c. 

12        1728 

9  6 

by  §64,     256  ^ H =  256,75578  &c. 

12         864 

by  the  decimals  above,  =  256,7556 

The  difference  of  nearly  two  units  in  the  fourth  decimal, 
or  the  ten  thousandth  parts,  is  owing  to  the  neglect  of 
the  last  decimals  of  the  factors,  which  of  course  gives  a 


DENOMINATE   FRACTIONS.  87 

smaller  result,  but  which  is  in  most  cases  sufficiently  ac- 
curate ;  a  farther  extension  of  the  decimals  would  of 
course  cause  a  nearer  approximation  to  the  other  result. 

§  66.  DIVISION  OF  DENOMINATE  FRAC- 
TIONS. The  remark  made  in  relation  to  the  mul- 
tiplication of  this  kind  of  fractions,  upon  the  incon- 
veniences of  the  operation,  and  its  being  applicable, 
generally  speaking,  to  lineal  dimensions  only,  ap- 
plies still  more  forcibly  to  their  division;  in  all 
other  cases  the  quotients  are  not  required  in  the 
same  denomination  of  subdivisions ;  and  in  cases 
where  the  divisor  and  the  dividend  are  of  a  dilFerent 
kind  of  quantity,  they  are  in  reality  impossible  in 
nature.  But  in  the  case  of  lineal  dimensions,  which 
produce  by  the  first  multiplication  superficial  mag^ 
nitudes,  and  in  a  second  solids,  as  both  objects 
really  exist  in  nature,  it  may  sometimes  be  desira- 
ble to  have  a  quotient  given  in  the  same  denominate 
fractions,  or  subdivisions. 

In  all  cases,  therefore,  where  such  a  division  oc- 
curs in  the  course  of  a  calculation,  the  nature  of  the 
quantities  concerned  are  left  out  of  consideration, 
and  the  quotient  inquired  into  is  considered  as  a 
mere  number.  To  do  this  two  ways  present  them- 
selves ;  either  to  reduce  every  different  denomination 
of  the  numbers  concerned  to  the  lowest  denomina- 
tion of  denominate  fractions  contained  in  them, 
and  divide  the  whole  numbers  resulting ;  the  quo- 
tient will  give  a  result  in  units  of  the  whole  ("not 
of  the  subdivision)  employed,  because  it  expresses  the 
value  of  the  general  fraction  expressed  by  the  divi- 
sion, which  is  itself  independent  of  the  subdivision 
employed  in  the  calculation. 

Or  the  denominate  fractions  may  be  reduced  into 
decimals  of  the  whole  quantity,  as  seen  in  the  pre- 
ceding §,  and  the  division  of  these  will  give  exactly 
the  same  result,  as  the  reduction  to  the  lowest  deno- 
mination ;  because  the  quotient  resulting  gives  also 


^^  DENOMINATE   ABACTIONS. 

liere  the  actual  value  of  the  fraction  expressed  by 
the  division. 

§  6r.  To  perform  a  Division  of  Denominate  Frac- 
tions, As  this  may  be  desired,  in  lineal  dimensions^ 
it  will  he  proper  to  give  an  appropriate  general 
rule ;  as  follows  : 

Find  the  whole  number  which,  multiplied  into  the 
divisor,  will  give  a  product  nearest  below  the  divi- 
dend, and  divide  by  it  as  in  common  division,  only 
minding  the  transfer  or  carrying  from  one  denomina- 
tion to  the  other,  according  to  the  principle  of  the  deno- 
ininate  fractioji;  then  reduce  the  remainder  to  the 
next  lower  denominatioii,  and  multiply  the  product 
by  the  denominator  of  the  denominate  fraction  ;  reduce 
also  the  whole  divisor  to  the  next  lower  denominate 
fraction,  and  divide  the  last  obtained  dividend  by  it; 
the  result  will  give  a  number  expressed  in  this  next 
lower  denomination ;  thus  continue  fo  the  end  of  all 
the  desired  subdivisions* 

The  reason  of  this  rule  is  evident  in  its  first  stej), 
from  common  division  of  whole  numhers ;  in  the 
second,  and  following  steps,  the  multiplication  of 
the  remainder,  after  reduction  by  the  denominator 
of  the  denominate  fraction,  is  necessary  to  give  by 
the  division  a  result  expressed  in  units  of  this  lower 
division ;  in  the  same  manner  as  for  decimal  frac- 
tions, a  0  was  added  to  every  remainder,  to  pro- 
duce a  quotient  of  the  next  lower  rank  of  decimals^ 
because  this  0  produced  a  multiplication  by  10. 

Example,     To  divide 
ft.  in. 
17^9^   =  2ft.  Sin.  +5\ 

7.  10 
15.    8 


^.    1  =  25  in.  multiplied  by  12 


DENOMINATE   TRACTIOKS.  89 

300  in.  12 

=  3  in.  +  — 

Divisor  reduced  =    94  94 

282 
12 

The  first  quotient  found  here,  or  the  feet,  is  2  ;  then  the 
remainder,  2  ft.  1  in.,  is  reduced,  and  multiplied  by  12,  to 
give  the  next  denominate  fraction,  by  the  division  with 
the  reduced  divisor,  or  94,  which  gives  3  in.,  and  the 
fraction  which  is  here  left,  but  could  be  reduced  again  to 
twelfth  parts,  as  the  next  subdivision,  by  the  same  opera- 
tion as  the  inches  were  obtained,  if  desired.  It  must  be 
observed  here  :  that  in  the  dividend  the  9  were  treated  as 
twelfth  parts  of  the  square  foot,  which  are  not  inches 
cubic  ;  if  they  were  such,  they  would  present  the  deno- 
minator 12  X  12  =  144,  as  may  easily  be  judged,  by 
reflecting  upon  the  multiplication  shown  above,  or  be- 
cause 1  ft.  =  12  in.  gives  1  ft.  square  =  12  in.  X  12  in. 
=  144  square  inches. 

To  execute  the  same  Operation  or  Division  btj  the 
two  other  Methods,  The  process  of  reduction  to  the 
lowest  denomination,  or  to  whole  and  decimal  num- 
bers,  is  evident  from  the  principles  of  division 
taught  for  the  two  cases,  in  their  proper  places. 

The  nbove  example  would  stand  in  them  as  follows  : 
1st.  By  reduction  to  the  lowest  denomination, 

ft.    in. 

17^%  213 

=  =  2,  2659  ft.  =  2  ft.  3, 1908  in. 

7.    10  94 

this  last  by  multiplying  the  decimals  by  12,  the  denomi- 
nator of  the  denominate  fraction. 

2d.  By  the  whole  numbers,  and  the  value  of  the  sub- 
divisions in  decimals, 

17-^  17  75 

— ^  =  — '- =  2,  2660  =  2  ft.  3,  192  in. 

7.  10         7,  8333 

le  first  result  will  be  somewhat  too  small,  on  account 

8# 


90  DEJSrOMINATB   FRACTIONS. 

of  the  discontinued  division,  the  second  somewhat  too 
large,  on  account  of  the  discontinued  series  of  3  in  the 
divisor,  which,  being  smaller,  leaves  the  quotient  to  be- 
come somewhat  larger. 

The  whole  of  the  examples  in  denominate  frac- 
tions, and  particularly  the  latter  ones,  show  :  that 
the  calculation  of  denominate  fractions  properly  be- 
longs to  the  applied  or  practical  part  of  arithmetic, 
which  is  intended  to  be  treated  in  the  next  chapter ; 
it  has  however  appeared  proper  to  treat  of  their 
principles  here,  considering  them  as  fractions  of  a 
particular  nature,  as  these  considerations  tend  to 
illustrate  the  general  view  of  fractions;  to  enter 
more  minutely  into  details  is  however  the  province 
of  the  practical  part,  where  more  examples  will  ap- 
pear, and  where  it  will  become  evident  to  every 
attentive  peruser  of  this  work :  that  the  proper  un- 
derstanding of  the  principles  of  arithmetic  will  sug- 
gest to  him  in  every  case  the  ideas  which  will  lead 
him  to  the  most  judicious,  accurate,  and  short  way 
to  execute  calculations,  implying  such  detail  cases. 


PART   II. 


PRACTICAL  APPLICATIONS  OF  THE  POUR  RULES   Oy 
ARITHMETIC. 


CHAPTER  I. 

General  Principles  of  the  Application  of  the  Four 
Rules  of  Arithmetic, 

§  68.  In  the  previous  chapters  have  been  deduced 
from  the  first  principles  of  the  combination  of  quan- 
tity and  numbers  what  are  called  the  Four  Rules  of 
Arithmetic,  and  they  have  been  applied  to  the  dif- 
ferent forms  in  which  quantities  are  presented, 
namely :  as  whole  numbers  of  units,  or  as  parts  of 
the  same ;  and  these  latter  expressed  either  by  their 
general  relation  to  the  unit,  as  in  Vulgar  Fractions, 
or  by  a  continuation  of  the  decimal  system  below 
the  unit,  as  in  Decimal  Fractions,  or  as  arbitrary 
subdivisions  under  the  name  of  Denominate  Frac- 
tions. 

The  preceding  part  may  therefore  be  considered 
as  the  theory  of  the  four  rules  of  arithmetic ;  it 
will  already  present  the  solution  of  a  great  num- 
ber of  the  questions  arising  in  common  life  from 
daily  intercourse  or  occupation.  Though  this  ap- 
plication might  be  made  by  the  teacher,  it  may  not 
be  improper,  particularly  for  such  persons  as  should 
wish  to  undertake  the  study  of  arithmetic  by  them- 
selves, to  give  a  few  leading  ideas  to  guide  them  in 
the  proper  choice  of  the  rule  for  a  certain  given 
case,  together  with  some  examples.  ^ 


9^  PEACTICAI.  APPLICATIONS   OP  THE 

5  69.  Under  the  head  of  Addition  will  come  :  all 
such  questions,  where  quantities  of  the  same  kind 
are  to  be  counted  together,  as  has  been  seen  to  be 
the  origin  of  this  first  rule.  It  is  of  course  impossi- 
ble, to  add  quantities  of  different  kinds  together  un- 
der any  denomination  than  as  mere  things ;  and  this 
remark,  simple  as  it  is,  may  escape  in  certain  cases. 
We  have  seen,  for  example,  in  fractions,  that  we 
were  compelled  to  make  such  changes  in  the  deno- 
minators as  produced  the  effect,  of  reducing  the 
quantities  which  are  of  a  different  kind,  on  account 
of  their  being  different  parts  of  the  unit,  to  quanti- 
ties of  the  same  kind,  or  denomination,  before  they 
could  be  added. 

That  all  tliis  applies  equally  to  Subtraction,  is 
evident  from  the  principle,  that  it  is  only  the  oppo- 
site operation  of  Addition. 

§  70.  So  for  example.  A  farmer,  making  the 
enumeration  of  his  live  stock,  may  add  them  either 
under  their  different  denominations,  as  different 
kinds,  or  in  sum  total. 

Suppose,  therefore,  a  farmer  had  his  live  stock  distri- 
buted in  different  lots  of  ground,  as  follows  : 

In  the  door  yard  are  3  cows,  each  with  a  calf,  2  horses, 
and  4  pigs. 

In  the  meadow  he  has  4  oxen,  6  cows,  and  3  young 
horses. 

In  the  field,  a  flock  of  35  sheep,  5  cows,  and  4  calves. 

He  lets  run  in  the  woods,  9  pigs,  7  cows,  4  young  oxen, 
and  2  horses. 

We  may  ask  here,  first,  the  sum  of  all  his  live  stock, 
which  will  comprehend  all  what  is  abot^e  under  one  sum  : 
thus  : 


FOLK  BUIES   OF  ARITHMETIC. 

2  horses 

3  cows 

3  calves 

4  pigs 
4  oxen 

6  cows 

3  horses 
35  sheep 

'  5  cows 

4  calves 
9  nigs 

7  cows 
4  oxen 
2  horses 


93 


Oxen. 

Calves. 

Horses. 

Pigs.   ; 

4 

8 

2 

4 

4 

4 

3 

2 

9 

8 

7 

13 

91   heads  of  live  stock. 
2d.  We  may  ask  how  many  of  each  kind,  and  then  we 
shall  have  to  separate  the  quantities   above,    in    this 
manner ; 

Cows.     Oxen.    Calves.   Horses.    Pigs,  i  Sheep. 
3  4  8  2  4  35 

6 
5 
7 

21 

Other  examples  of  Simple  Addition  may  be  the  fol  • 
lowing  : 

For  a  certain  undertaking  in  a  village,  seven  men 
agree  to  give  each  as  much  money  as  he  has  in  cash  in 
pocket;  John  has  ^47;  Peter  gl21;  James  g30 ;  Rich- 
ard g79  ;  Francis  gl07  ;  Frederic  gl92  ;  and  William 
J305  ;  how  much  stock  do  they  bring  together  ?  The 
addition  gives  :  $  47 

121 
60 
79 
107 
192 
305 


^901 


94      PRACTICAX  APPIICATIONS  OP  THE 

How  much  is  the  whole  banking  stock  in  New-York, 
(he  stocks  of  the  chartered  banks  being  as  follows 
Bank  of  New- York,  $     930, 000 

Manhattan  Bank,  2,  050,  000 

Merchants'  Bank,  1,  490, 000 

Mechanics'  Bank,  2,  000,  000 

Union  Bank,  1,000,000 

Bank  of  America,  2, 000,  000 

City  Bank,  2,  000,  000 

Phoenix  Bank,  600, 000 

United  States'  Bank,a      35, 000,  000 
Frankhn  Bank,  500,000 

North  River  Bank,  500,000 

Tradesmen's  Bank,  600, 000 

Chemical  Bank,  600, 000 

Fulton  Bank,  600,  000 

Examples  of  this  kind  are  too  easy  to  require  the  inser- 
tion of  more. 

§  71.  Examples  of  Subtraction. 

1.  Francis  has  35  head  of  cattle  on  his  farm,  and  his 
neighbour  James  84  ;  how  much  has  the  one  more  than 
the  other? 

James'  cattle         84 
Francis'  cattle      36 

Difference  49     which  James  has  more. 

2.  A  man  going  into  account  with  himself  finds  his 
whole  property  amounts  to  J 18406,  and  that  he  has 
.$10509  debts  ;  how  does  he  stand  ? 

Property         ^  18406 
Debts  10509 


Difference       ^  7897     clear  property  left. 

3.  In  a  year  there  are  365  days  ;  of  these  62  are  Sun- 
^lays ;  how  many  working  days  are  there  in  a  year  ? 

Ans.  313  days. 

4.  A  man  in  business  bought,  during  the  year,  goods  to 
the  jynount  of  ^106409,   and  sold  to  Jhe   amount  of 


FOUR  BUIES    OF  ARITHMETIC.  95 

j?59879  ;  taken  at  the  same   price   or  estimate,   what 
amount  of  goods  has  he  left  ?  Ans.  §46530. 

§  72.  The  applications  of  Multiplication  occur  in 
every  case  where  one  of  the  quantities  occurs  as 
often  as  indicated  by  another  number,  which  forms 
the  multiplier,  as  is  the  case,  for  instance,  in  all 
purchases,  profits,  interest,  at  a  certain  rate  for  the 
adopted  unit  of  the  things  bought,  sold  or  lent,  or 
in  general,  whenever  the  same  thing  or  quantity  is 
repeatedly  taken. 

1st  Example.  John  buys  12  peaches  at  3  cents  a-piece  ; 
how  much  has  he  to  pay  ?  Ans.  36  cents. 

2d.  48  head  of  poultry  bought  at  6  cents  per  head  ; 
how  many  cents  to  pay  ?  Ans.  240. 

3d.  The  yefg:  has  365  days,  every  day  24  hours  ;  how 
many  hours  in  a  year  ?  Ans.  8760  hours. 

4th.  How  many  minutes  are  there  in  a  month  of  31 
days,  the  day  having  24  hours,  and  the  hour  60  minutes  ? 

Ans.  44640  minutes. 

6th.  A  merchant  bought  56  bales  of  cotton  goods;  15 
of  them  held  21  pieces,  29  held  28  pieces,  and  the  rest 
25  pieces  each  ;  for  each  piece  he  pays  g3  ;  how  much 
must  he  pay  for  the  whole  ? 

^^^J  $4281  to  pay. 

I  1427  pieces  of  cotton  goods. 

The  numbers  indicating  the  quantity  of  pieces  in  each 
bale  are  to  be  multiplied  by  the  number  of  bales  respect- 
ively;  the  sum  of  these  results  gives  the  whole  number 
of  pieces,  which  being  multiplied  by  3,  the  price  of  each 
piece,  gives  the  final  result. 

6th.  A  merchant  bought  963  barrels  of  flour  ;  on 
weighing  them,  he  finds  their  average  weight  202  lb. 
and  that  the  barrels  average  7  lb.  weight  each  ;  how 
many  pounds  of  flour  has  he  ?  Ans.   187,785  lbs. 

Note.  The  subtraction  needed  in  tbe  above  example 
from  each  barrel,  is  what  in  commerce  is  called  tare  ; 
the  remaining  weight  is  what  is  called  neat  weight.  Tare 
is  usually  determined  by  an  approximate  valuation,  in 
each  particular  kind  of  package,  according  to  certain  ha- 
bitual, and  even  local,  rules.     It  is  sufficient  to  know 


96     PRACTICAL  APPIICATIOXS  OF  THE 

these,  to  execute  any  example  of  mercantile  calculation 
relating  to  what  is  called  tare,  as  they  form  a  subtraction 
upon  agreed  principles. 

7th.  The  sum  of  ^6500  is  lent  out  at  interest,  for  three 
years,  at  6  per  cent,  simple  interest,  annually  ;  what 
will  be  the  whole  amount  of  that  interest  in  three  years  ? 

The  interest  per  cent  being  evidently  a  decimal  frac- 
tion, in  the  place  of  |the  hundreds,  or  second  decimal ; 
the  whole  operation  of  any  interest  calculation  for  the 
year,  consists  in  multiplying  the  capital  given  with  the 
corresponding  decimal  fraction,  and  for  more  years,  to 
multiply  this  product  again  by  the  number  of  years 
required  ;  thus  the  above  consists  in  the  execution  oi 
the  following  multiplication : 

6500  X  0,06  X  3  =  390  X  3  =  gll70. 

It  is  evident  also  from  this,  that  all  transactions  of 
commission,  brokerage,  exchange,  notes,  drafts,  stock, 
&c.,  which  are  grounded  upon  a  certain  per  centage  of 
premium,  or  discount,  are  exactly  of  the  same  nature, 
and  determine  a  decimal  fraction  by  which  the  amount 
is  to  be  multiplied,  as  in  this  example. 

5  72.  Division  applies  in  ordinary  business  to  all 
cases,  where  any  quantity  of  things  is  to  he  divided 
among  an  equal  number  of  persons,  or  in  an  equal 
number  of  lots  or  parts ;  the  quotient  will  give  the 
share  of  each  person,  or  the  quantity  of  things  in 
each  lot  or  part.  It  will  therefore  also  apply  to 
find  the  price  of  a  single  piece  of  a  thing,  of  which^ 
a  large  number  has  been  purchased,  for  a  certain 
price ;  as  in  the  following  examples  : 

1st  A  father,  having  six  sons,  leaves  among  them  a 
property  of  §76590,  to  be  shared  equally  among  them  j 
how  much  will  each  son  get?  Ans.  g  12765. 

2d.  The  provision  of  an  army  in  bread  is  90567  lb. : 
it  is  intended  to  distribute  the  whole  to  the  soldiers,  to 
save  separate  transportation  ;  there  are  10063  soldiers  j 
how  many  pounds  will  each  have  to  carry  ?     Ans.  9  lb. 

3d.  The  expenses  of  paving  a  street,  500  feet  in  length, 
amount  to  ^1000  ;  the  amount  is  to  be  distributed  among 


rOUR  BUIES    OF   ARITHMETIC.  97 

the  owners  ol"  the  Hcijoining  lots,  each  having  a  lot  of  25 
feet ;  how  much  wili  each  lot,  or  owner,  have  to  pay  ? 

Ans.  g50. 

4th.  A  merchant  bought  109  bales  of  calico,  for  ihe 
total  amount  of  $12232  ;  he  finds  that  40  bales  contain 
each  30  pieces,  60  contain  each  25  pieces,  and  the  rest 
contain  32  pieces  each  ;  how  high  does  each  piece  stand 
him  ?  Ans.  $4. 

5th.  How  many  days  are  there  in  24480  minutes,  each 
day  having;  24  hours,  eaf  h  hour  60  minutes  ?   Ans.    7ds. 

6th.  How  many  days  will  it  take  a  man  to  travel  946 
miles,  if  he  travel  .36  miles  per  day  ?  Ans.   27  ds. 

Upon  the  principle!^  here  shown  for  the  four  rules  in 
xvhole  numbers,  it  will  be  easy  for  the  teacher  and  scholar 
to  form  abundance  of  examples  for  practice,  and  to  solve 
those  given  at  the  end. 

§  73.  It  will  be  proper  here  to  draw  the  attention 
to  a  general  principle,  which  will  always  guide 
ns  in  the  use  of  multiplication,  as  applied  to  any 
purpose  of  life,  or  even  of  science;  namely  :  it  ex- 
presses  always  by  the  two  factors  a  certain  cause 
and  a  certain  repetition  of  the  same,  which  may 
be  best  represented  by  time,  for  this  is  the  measure 
of  repetition  of  effects  in  nature,  as  we  have  seen  it 
to  be,  for  instance,  in  calculations  of  interest,  &c.  | 
the  result  of  these  factors,  giving  the  product  of  the 
numbers,  represents  e  jually  well  the  effect  produced 
by  the  cause,  represented  by  the  one  factor  acting  a 
certain  time,  which  is  repi-esented  by  the  other  factor. 

So  we  may  represent  the  multiplication  and  its 
results  as  the  product  of  cause  into  time,  being  equal 
to  their  effects  (the  great  law  which  is  to  be  exactly 
filled  in  every  explanation  or  investigation  of  a  sub- 
ject of  natural  philosophy.)  Instead  of  time,  we 
may  also  call  that  factor  power,  and  the  other  the 
object  acted  upon  by  this  power ;  the  ideas  of  cause 
anti  time  however,  always  apply  equally  well ;  as  for 
instance,  a  man  having  certain  means  to  do  a  thing. 

.9 


98     PRACTICAL  APPLICATIONS  OF  THE 

and  using  them  so  much,  or  so  many  times,  would 
be  the  same  thing  in  respect  to  the  effect;  and  so 
in  all  similar  cases. 

As  we  have  shown  in  multiplication,  that  its 
application  included  all  the  cases  where  a  cause 
acting  a  certain  time,  or  number  of  times,  pro- 
duced a  certain  effect,  So  division  may,  with 
equal  propriety,  be  considered  as  decomposing  the 
effectf  Mo  its  cause  and  time;  the  one  of  these 
being  given,  besides  the  effect.  Thus  we  evidently 
find:  that  if  a  certain  work  had  been  done,  by  a 
certain  number  of  men,  in  a  certain  time,  the  work 
expressed  in  numbers  representing  the  effect,  tlie 
time  of  the  work,  or  the  number  of  men,  being  giv- 
en, the  number  of  men,  or  the  time  of  their  work, 
may  be  found,  by  dividing  the  effect  by  the  number 
of  men,  or  the  time  they  worked.  In  like  manner: 
if  the  interest  btaintd  upon  a  sum  of  money  in  a 
certain  number  of  yeas  be  given,  the  yearly  amount 
of  it  (that  is  the  cause)  will  be  given,  by  the  quotient 
arising  from  the  division  of  the  w  hole  amuunt  by 
the  number  of  years,  and  vice  versa. 


CHAPTER  II. 

Application  of  the  Four  Kules  of  Jlrithmetic  to  all 
iiinds  of  ^uestions^  iuvoLviug  Fractions  of  either 
kind, 

§  74.  In  most  of  the  circumstances,  where  calcula- 
tion is  to  be  applied  in  common  lite,  the  given  quan- 
tities either  contain  certain  fractions,  or  denominate 
subdivisionsof  the  unit,  which  have  been  called  De- 
nominate" Fractions,  or  they  often  lead  to  such  by 
division,  as  has  been  seen  in  its  place.  The  calcu- 
lator must  determine  by  tlie  aid  of  pj-oper  reflection 
upon  any  given  case,  and  by  liis  knowledge  of  the 
principles  or  theory  of  arithmetic,  in  what  manner 


FOim   RULE 3    OF   ARITHMETIC.  99 

it  will  be  most  easy,  and,  according  to  the  aim  of 
the  calculation,  most  accurate,  to  obtain  the  result. 
Practice  gives  great  facilities  for  this  determina- 
tion;  in  the  instructions  lor  performing  it,  only 
general  considerations,  or  priniriples,  can  be  pre- 
sented, and  examples  that  may  serve  as  an  intro- 
duction to  it ;  this  is  the  aim  of  the  present  chapter. 

It  may,  for  instance,  be  reatiily  inferred  from  a 
comparison  of  the  operations  in  Vulgar  and  in  De- 
cimal Fractions,  that  in  complicated  additixms  of 
numbers,  involvitig  vulgar  fractiojis  whose  denomi- 
nators are  not  simple,  or  commensurable,  (that  is, 
the  one  a  multipl(*  of  the  other,)  the  reduction  of  the 
fractional  part  into  decimal  fractions,  before  they 
are  added,  is  peculiarly  advantageous. 

In  subtraction  the  same  is  the  case,  in  a  less  de- 
gree however,  on  account  of  the  circumstance  of 
there  being  only  two  fractions  that  can  possibly  be 
engaged  in  one  operation. 

Denominate  fractions  present  no  difficulty,  in 
either  addition  or  subtraction^  more  than  common 
numbers,  except  the  attention  that  is  necessary  in 
the  carrying,  or  borrowing,  from  one  denomination 
to  the  other,  but  are  from  that  circumvStance,  and 
their  irregular  progressions,  far  less  convenient 
than  decimal  fractions.  From  this  circumstance  the 
decimal  system  derives  great  advantage,  and  has 
for  that  reason  been  introduced  at  least  in  the  coins 
of  the  United  States. 

The  reduction  of  whole  numbers  into  fractions 
will  never  be  needed  in  addition;  when  it  may  be 
required  in  subtraction,  the  application  of  the  prin- 
ciple of  borrowing  one  single  unit,  and  reducing 
the  same  into  the  required  fraction,  as  in  the  addi- 
tion of  whole  numbers  and  decimals,  will  be  the 
most  advantageous  and  shortest  method,  wherever 
4he  reduction  of  the  vulgar  fraction  into  a  decimal 
fraction  does  not  present  greater  advantages. 


100     PRACTICAL  APPLICATIONS  OP  THE 

The  preceding  remarks,  in  addition  to  what  has 
been  said  upon  the  application  of  the  two  first  rules 
of  arithmetic,  may  suffice  in  this  place ;  particu- 
larly as  in  multiplication  and  division  they  again 
naturally  occur,  and  receive  their  explanation  and 
application,  in  a  manner  still  more  instructive,  than 
when  treated  alone. 

A  few  examples  placed  here  for  exercise  may  therefore 
suffice  ;  they  will  be  expressed  by  the  signs  of  the  ope- 
ration, when  eiven  simply,  in  order  at  the  same  time  tt 
afford  an  opportunity  of  exercise  in  their  use. 

1st.  Execute  =17-1-4+  V  +3  +  ^+9+1  = 
2d.      "     I  4-1  +5-4- ^^2+^  +  }^  +  6  +  ^  = 
3d.      "     tV-  H-3  +  1  +  7  -  f.+  6-^\  = 

5th.     ''     4,65080906  +  0,0070606  +  1 14,604091 

4-  0,985  4-  406,307506  +  3000,040907  = 
6th.     "     6,04097062  -  5,908986072  = 
7th.     "     3,4091  -  3,064723  -f-  5,08(»9701  -  2,90806£ 

-I-  101 ,01980-67.520998  +  3,05  -  0,0672  = 
3lb.     "     13  ft.  7,5  in.  -|-  4  ft.  6,3  in.  +  16  ft.  0,5  in. 

4-  0  ft.  7,13  in.  = 
9th.     *'     6  ft.  9,2  in.  -}-  0  ft.  1 1 ,6  in.  -  6  ft.  4,25  in.  = 
10th    "     3lb.  7oz.  2dwt.  7gr. -|- 6oz.  7dwt.  19gr.' = 
nth    "     107  lb.  4  oz.  0  dwt.  6  gr    -f  7  oz.  6  dwt.  2  gr. 

4-  5  dwt.  19  gr   4  6  lb   0  oz.  7  gr.  -  106  lb. 

10  oz.  16  dwt.  20  gr.  = 
Question  1 .   A  farmer  thrashed  grain  seven  days  ; 
the   1st  day    12|  bushels 


»> 

2d 

18i 

;' 

3d 

241 

»j 

4th 

30f 

it 

5th 

32f 

a 

6th 

44| 

5) 

7th 

15i 

He  paid  his  help  in  grain  ;  to  one  man  he  gave  3i  busk 


y^  W  i^^ 


FOUR   RULES   OF   ARITJIWTIS*  101 

lo  another  2^  ;  and  he  returned  to  t;is.neighboar  vh^t  h<} 
had  last  borrowed  of  hira  to  go  tc  rriHlVwhich ''■Vas  !J|. 
bushels  ;  how  much  grain  has  he  left  ?  Ans.  IGSy^jbush. 
2d.  A  man  goes  out  to  collect  payment  of  bills  ;  he 
pays  also  his  own  debts  in  going  his  round  ;  thus  he  gets 
from  James  g77,  65  ;  from  William  gl05,  37|;  then  he 
pays  his  grocer  ^98,  12^  ;  going  on  he  gets  payment  from 
P.  Jones  ^3U7, 62i  ;  and  from  J.  Johnson  he  gets 
g692,  875  (=  87|  cts.  ;)  now  he  thinks  himself  able  to 
make  a  payment  on  his  house  of  (with  interest)  ^856,  626, 
and  pays  his  tailor  yet^28,  375  ;  how  much  has  he  left 
when  he  gets  home?  Ans.  §200,4 

§  7b,    We  have  seen  that  the  multiplication  of 
whole  numbers  alone,  and  of  fractions  alone,  pre- 
sents no  diihculty,  while  the  mixture  of  both,  as  we 
have  given  an  example,  by  reducing  the  denominate 
fractions  to  fractional  parts,  with  small  denomina- 
tors, equivalent  to  them,  has  shown  an  operation, 
which  we  would  gladly  have  exchanged  for  one  on 
the  same  principle  as  the  decimal  system.     Still  it 
will  always  depend  on  the  judgment  of  the  calcula- 
tor, to  which  mode  he  shall  give  the  preference,  if 
his  data  are  partly  given  in  fractions ;  because  these 
are  often,  even  more  generally,  in  no  complicated 
proportions,  as  for  instance,  i;  \j  {\  i;  |^  =; 
and  the  like;  and  particularly  when  only  one  factor 
lias  a  fraction,  the  operation  may  be  easily  enough 
performed,  to  permit  the  like  reductions  to  be  avoid- 
ed, which,  may,  for  instance,  in  J;  ^;  |;  and  the 
like,  lead  to  interminate  decimals.  In  this,  therefore, 
the  judgment  of  the  calculator  must  decide,  and  it 
is  very   improper  to  bind  one's  self  to  any  single 
peculiar  mode  5  reflection  will  lead  to  a  calculation 
easy  and  accurate,  w  hile  a  mere  mechanical  process 
will,  wlien  a  mistake  occurs,  cause  embarrassment. 
What  we  would  here  advise  is,  good  order  in  all 
calculations;  that  any  example,  however  compU- 
9# 


(02.  PHACTICAX-  APPLICATIONS  OP  THE 

c&ted^  .be  written . distinctly  and  regularly,  in  the 
order-  in-  which  it  proceeds,  accompanied  by  the 
signs  of  the  operations  that  are  appr<»priate,  whene- 
ver the  operation  itself  might  not  declare  it  dis- 
tinctly. All  this  is  nothing  else  but  the  necessary 
and  well-known  principle,  that  every  thing  must  be 
done  with  reflection  and  order,  if  it  is  to  succeed. 

The  mode  of  proceeding  will  most  likely  be  best  elu- 
cidated by  a  few  examples,  of  different  kinds,  accompa- 
nied by  appropriate  reasoning. 

1st  Example.  Seventeen  packages  of  goods,  each 
weighing  72^  pounds;  what  is  the  total  weight  ? 

Ans.   1231  lb, 

2d.  If  one  hundred  weight  of  wool  is  bought  at  4U.\^ 
•lollars,  what  will  17f  hundreds  cost? 

Write  this  example  thus  ; 


17. 

6 
4 

40. 

i 

»_ «. 

m-m 

680. 

30. 

5. 

H 

716.  H 

The  first  product  is  obtained  by  the  multiplication  of 
17  X  40;  then  f  x  40  gives  the  second  line;  then  mul- 
tiplying the  17|  by  i,  or  what  is  the  sanje  thing,  taking 
the  third  part  of  it,  the  17  gives  6.  and  the  remaining 
whole  quantity  2  ;  which  reduced  to  fourth  parts,  gives  | 
to  be  added  to  the  fraction  f,  jiiving  y ;  which  are  to  be' 
divided  by  3  ;  and  give  the  fractional  part  ji  ;  the  sum 
of  all  the  three  products,  is  the  product  of  the  whole 
numbers,  and  fractions,  into  each  other,  as  required. 

3d.    To  give   an  example  requiring  more  fractional 
operation,  let  the  following  multiplication  be  given  : 


^V  I-OVA   RULES    OF   ARITHMETIC.  lOS 

B  34. 1 

following  the  operations  as  before,  (the  opera-        19.  | 

tioDS  shall  here  be  denoted  by  the  signs.) . 

9  X  34  =  306 
10  X  34  =  340 
Decomposing      \  (1^X^=9.  i 

the  fraction  I     \  f  X  19  =  ^  19  x  i  =      4.  f 

and  the  fractiot.  ^  «  v  r.^d  O.  1^  =  )  xhl ^ml  =      e!  || 


[id  the  fraction  ?  »  . .  /«^    ,    ,\  )  ,u^^ 

intoX  +  i  +  xl?X(34  4.,)-j.he 


tinioii-i-rt>^       ^  •'        <thesarae=      6,  || 

341  X  19f  =  683.11 

In  the  same  way  any  other  example  of  mixed  numberi 
would  be  carried  on  ;  this,  which  was  chosen  to  show 
how  to  decompose  the  fractions  for  the  operation,  would 
evidently  not  be  executed  in  this  manner  by  a  reflecting 
calculator;  for  the  reduction  to  decimals  is  easy, and  ifte 
decimal  fractions  terminate  ;  thus  : 

34^  =     34,875 
793  ^     19,6 

313,875 
348,  75 
20,  9250 


683, 5500 


is  multiplied  with  the  greatest  ease,  and  reducing  back- 

55  11 

wards  again  ;   the =  — ,  by  reduction  by   5, 

100         20 

5 

55  ^1 1 


100 


20 


4th.  A  lumber  merchant  has  12  pieces  of  timber  of 
461  feet  long,  and  15  inches  square  ;  how  many  cubic 
feet  has  he  ;  and  what  sum  total  will  he  get  for  it  at  I81 
cents  the  cubic  foot  ?  >      - 


104 


PRACTICAI.  APPLICATIONS  0¥  THE 


Here  we  may  evidently  P'  oceed  either  by  Denominate 
Fractions,  and  Practice,  or  Cross  Multiplication,  or  by 
Decimals  ;  for  the  latter  the  exwmple  presents  the  great- 
est facility,  on  accouni  ot  the  fractions  being  easily  redu- 
cible ;  it  shall  be  made  nere  in  the  three  ways,  parallel  to 
each  other,  for  the  contents  of  each  piece,  the  rest  being 
in  simple  quantities,  and  the  ruoney  of  decimal  division 
is  in  all  cases  best  suited  for  decimal  multiplication. 


By  Cross 
Multiplication. 
46.    6 


1. 

3 

11. 

46. 

7. 
6 

6 

58. 
1. 

1. 
3 

6 

14. 

58. 

6. 

4. 

6 

6 

72.    7.  10.    6 


By  Deci- 

mah. 
1,25 
1,25 


1. 

25 

250 

625 

1, 

5625 

46, 

5 

% 

3750 

62, 

600 

78125 

By  Practice. 
ft.    in. 
1.    3 
1.    3 


1. 

3 

-1 

1. 

46. 

6 

46. 
23. 

1. 

0. 

6    in.   ft. 
3  =6  =  i 

72. 

n 

2  '^2  4 


72,65625 

The  cubic  coritents  of  each  piece  being,  in  decimal 
fractions,  72,65625  feet,  the  12  pieces  give  871,875 
cubic  feet,  and  thet?e,  at  I2i  cents  or  0,  125  dollars,  bring 
the  amount  by  multiplying  871,  874  x  0,125,  or,  what 
is  the  same  thing,  87  I, "^874  X  ^  =  108,  984375  dollars  ; 
for  which  would  in  actual  practice  be  given, 

Ans.  $108,  98i 

^7&.  The  Division  of  quantities,  expressed  in 
whole  numbers  aud  fraction;d  parts,  either  vulgar 
or  denominate,  when  the  dividend  only  has  fractions, 
ran  be  made  as  in  common  numbers  ,*  after  having 
divided  the  whole  number,  the  remainder  is  reduced 
into  a  fraction  of  the  same  denominator  as  tke 
fraction  given,  and  the  fractional  part  jbeing  added, 


I 


VOVn  RULES    OP   AKITHMETIC.  105 

the  division  is  continued,  and  gives  fractions  of  the 
same  denominator;  so  may  be  continued  as  far  as- 
desired  for  the  intended  aim. 

For  decimal  fractions,  the  directions  given  §  58 
may  suffice. 

When  both  dividend  and  divisor  aie  numbers 
mixed  with  fractions,  it  will  be  found  tlie  most 
satisfactory,  tlierefore  most  generally  the  easiest, 
to  reduce  the  whole  numbers  of  both  to  the  denomi- 
nator of  the  fraction  annexed  to  each ;  and  wri  ing 
the  resulting  fractions,  execute  the  division  as  shown 
in  its  place. 

When  the  divisor  and  dividend  have  both  deno- 
minate fractions,  we  have  seen  (§  67)  that  the  divi- 
sor in  that  shape  is  unwieldy  and  disadvantageous, 
and  therefore  we  have  there  shown  two  methods 
applicable  with  nearly  equal  advantages,  to  which 
"we  therefore  refer. 

In  the  practical  application,  therefore,  either  the 
one  or  the  other  of  these  modes  will  be  cho- 
sen, according  as  it  presents  t  e  greatest  advanta- 
ges ;  for  it  is  evidently  useless  to  raise  difficulties 
in  a  practi<  al  work  of  any  kind,  to  have  the  plea- 
sure or  glory  of  solving  them.  The  object  here 
will  therefoi'C  only  be  :  to  present  such  questions 
as  are  soluble  by  division,  either  alone,  or  com- 
bined with  the  preceding  rules ;  for  it  must  have 
been  observed,  that  it  is  a  general  principle  in 
arithmetic :  t.'<at  all  the  preceding  rules  are  applied 
in  any  subsequent  one;  so  in  practical  questions 
the  same  liberty  must  be  allowed,  and  it  is  proper 
always  to  sh«)W,  in  any  stage  of  a  science,  what 
can  be  done  with  what  has  been  taught,  up  to  the 
step  which  is  making. 

1st  Examptt.  T<i  exefcute  the  division by  reddc- 

ing  the  whole  numbers  to  improper  fractions,  and  re- 
ducing the  result,  thus  : 


^€6  l»«ACTICiL  APPLICATIONS  ©:»  THE 

13X5-f4         69 


6  X  8  -f-  7 


5           69  X  8 

552 
275 

276 

55         55  X  5 

8  8 

This  operation,  step  for  step,  is  sufficiently  clear  from 
the  principles  effractions  ;  the  reduction  of  the  whole 
Dumbers  into  fr  ctions.  with  the  aildition  of  the  numera- 
tors, gives  a  fraction  to  be  divided  by  another,  and  naul- 
tipljing  hoth  the  nunnerntor  and  denominator  of  this 
compound  fraction  by  hoth  dtnominators  successively, 
the  two  whole  numbers  result,  which  are  to  be  divided 
for  the  fitial  result  The  application  of  this  principle 
has  no  other  difficulty  in  nrtore  complicated  numbers,  than 
the  length  of  th?  multiplications  ,  this  example  may 
therefore  suffice. 
In  decimal  fractions  this  example  would  st,and  as  foUows  :. 
13,8 

=.  2,00727272  &c. 

6,875 

354f 
-•2^4    To  execute  - — —  by  both  method. 

84^ 

3d.     To  execute -by  both  methods. 

62i| 

4tfc.  A  farmer  has  mowed  78i  acres  df  meadow  land, 
v^hich  yielded  on  an  average  2|  tons  per  acre,  and  be- 
sides having  wintered  with  the  hay  63  head  of  cattle,  he 
has  sold  62J-tons;  at  what  average  per  head  has  his 
cattle  consumed  the  hay?  -  Ans.  2i  tons. 

•  And  how  much  hay  did  he  make  in  the  whole  ? 

,     Ans.  209|  tons. 

5th.  Six  men  undertake  to  make  the  hay  on  a  piece 
of  land  for  65  dollars,  besides  their  board,  which  they  do 
in  1 1  days,  and  make  39  tons  of  hay  ;  at  what  rate  did 
(hey  make  wages,  how  much  per  ton  did  the  hay  cost  te 


FOIJB   RUI.ES    OP    ARITHMETIC.  lOT 

UQuke,  reckoning  the  board  at  25  cents  per  day  for  eacb 
man,  and  how  much  hay  did  each  man  make  on  an  ave- 
rage per  day  ? 

They  made  i|  of  a  ton,  or  half  a  ton  and  j\  per  day. 
'*         "     981  cents  per  day,  nearly,  besides  board. 
The  hay  cost  g2, 09  nearly  per  ton  to  make. 
6.    A  man  undertakes  a  job  for  §195;  he  hires  for 
help  5  men,  at  the   rale  of  ty2^  cents  per  day  and  their 
board,  which  he  calctilates  to  cost  him  for  each  25  cents 
daily  ;  he  works,  with  these  men,  30  days ;  how  much 
wages  did  he  make  per  day,  paying  his  board  at  the  same 
rate  as  the  men  he  hired  ?  Ans.  §1,  875  per  day. 

And  how  much  did  his  men  cost  him  ?  Ans.  §131,  25 
7th.  A  sura  of  §  1 3 1 1  is  to  be  paid  in  six  equal  instal- 
ments, with  interest  at  7  per  cent,  each  time  upon  the 
sum  unpaid,  the  first  injstalment  being  on  the  delivery  of 
the  goods,  the  others  yearly.  What  will  each  payment 
amount  to  ? 

The    1st  will  be  §218,5 
"       2d  "  294,975 

"       3d  "  279,68 

"      4th        "  264,385 

•'       5th        "  249,09 

"       6th        "         233,795 

Dividing  the  whole  sura  by  6  gives*  the  equal  yearly 
payments  of  the  capital,  which  subtracted  every  year 
from  the  last  capital,  giv<^s  the  capital,  the  interest  of 
which  at  7  per  cerit.  is  to  be  added  e  i.:h  year  to  the 
equal  payments,  the  first  or  present  payment  having  no 
interest  upon  it. 

8th.  There  are  in  a  page  of  this  book  on  an  average 
2000  pieces  of  type  ;  the  space  filled  by  them  is  3,  I 
inches  broad  and  5,  b  inches  long  ;  what  square  space  in 
;•,  mean  does  each  piece  occupy  ?  An^.  n  fwino^r.. 


PART   III. 

OF   RATIOS    AND    PROPORTIONS. 

CPAPTER  I. 

Elementary  Considerations  of.  Ratio. 

§  77.  In  the  very  outset  we  have  shown,  that 
quantity  was  all  that  is  capable  of  increase  or 
decrease,  without  regard  to  tiie  nature  or  kind  of 
things  the  number  of  which  was  increased  or  de- 
creased. From  the  simple  step  of  considering  two 
things  togeth<T,  or  adding  them,  and  then  succes- 
sively more,  or  diministiing  a  certain  number  of 
things,  both  by  the  use  of  a  determined  system  of  nu- 
meration; we  have  arrived  step  by  step  at  tlie  prin- 
ciples of  the  combination  of  quantity,  and  conversely 
again,  to  tlie  decomposition  of  a  combination  into  its 
parts. 

This  process  has  led  us  to  the  foiir  rules  of  arith- 
metic, that  have  been  explained  successively,  two 
of  which  have  been  shown  to  be  the  opposite  of  the 
two  others;  each  leading  alter  ately  to  the  decom- 
position of  the  composition  of  the  other,  as  addition 
and  subtraction,  multiplication  and  division ;  the 
latter  two  of  winch  have  been  show  n  to  be  the  result 
of  tlie  continued  repetition  of  the  two  first.  By  these 
means  all  the  operations  upon  quantity  in  usual  life, 
which  depend  merely  on  combinations,  have  become 
calculable,  as  shown  by  the  application  made  of  this 
theory  in  the  second  part. 

This  retrospective  view  of  the  part  of  arithmetic 


ELEMENTARY  CONSIDERATIONS  OF  RATIO.    109 

hitherto  treated  of,  appears  proper  to  betaken  here, 
in  order  to  awaken  appropriate  reflections  in  refer- 
ence to  the  whole  of  what  has  been  done,  and  the 
means  it  has  furnished  for  further  progress.  The 
scholar,  atten-  tive  to  what  he  has  done  hithei-to, 
cannot  but  have  acquired  the  faculty  of  reasoning 
upon  quantity.  The  reflections  which  we  shall  have 
to  make  in  future  will  be  as  simple  as  before,  but 
the  application  of  them  will  require  that  he  have 
made  himself  acquainted  with  the  tools,  or  means, 
which  he  has  to  use  in  the  following  parts  of  arith- 
metic, andacquired  some  dexterity  in  their  use ;  he 
will  do  well  therefore  to  cast  back  upon  the  whole 
a  cursory  view,  in  order  the  better  to  comprehend 
the  general  ideas  that  have  directed  it. 

§  7S  The  consideration  which  will  he  the  foun- 
dation of  the  part  of  arithmetic  to  be  now  treated  of, 
is  the  relation  which  the  quantities  may  have  to  each 
other,  whether  they  be  combined  in  any  way,  or 
not. 

The  relation  of  quantities  to  each  other,  in  what- 
ever way  it  may  be,  is  called  their  ratio.  As  we 
have  seen  that  the  increase  or  decrease  of  quantities 
depends  on  their  combination,  so  their  relation  to  each 
other,  that  is,  their  ratio,  must  also  depend  on  their 
possible  combination,  as  it  is  determined  by  it.  The 
ratio  is,  therefore,  also  considered  in  relation  to  these 
combinations  ;  and,  as  we  have  had  the  two  princi- 
pal combinations,  of  addition  or  subtraction,  and  of 
multiplication  or  division,  so  we  have  also  two  kinds 
f)f  ratio,  corresponding  to  them ;  namely,  by  addi- 
tion or  subtraction,  and  this  is  called  arithmetical 
ratio ;  and  by  multiplication  or  division,  which  is 
called  the  geometrical  ratio.  We  evidently  here  again 
Bnd  the  second  a  repetition  of  the  first,  as  multipli- 
cation and  division  are  the  repetition  of  addition  and 
subtraction ;  but  we  may  omit  going  so  far  back 
7nto  elementary  considerations,  and  proceed  forward 
10 


110    ELEMENTARY  CONSIDERATIONS  OF  BATIO. 

with  the  general  idea,  to  render  it  fruittul  for  prac- 
tical use.* 

These  two  kinds  of  ratio  take  their  mark  of  nota- 
tion from  tiie  marks  applied  to  the  combinations  or 
rules  of  arithmetic,  on  which  they  depend  ;  thus  : 

The  arithmetical  ratio  of  7  to  3,  is  expressed  by 
7  —  3 

The  geometrical  ratio  of  7  to  3,  is  expressed  by 

7 
7  :  S  ;  or  — 
3 
They  might  be  equally  well  expressed  by  the  signs 
of  addition  and  multiplication,  if  we  were  in  the 
habit  of  generalizing  the  considerations  on  quantity 
to  that  extent ;  and  we  shall  see  hereafter,  that  their 
theory  leads  to  it ;  that  is  to  say,  that  when  the 
ratio  of  two  quantities  by  subtraction,  or  division, 
to  which  the  above  signs  are  appropriated,  are  giv- 
en, their  ratio  by  addition,  or  multiplication,  is  also 
given ;  or  the  one  is  a  consequence  of  the  other.  In 
the  habitual  mode  of  writing,  therefore,  an  arith- 
metical ratio  expresses  a  difference  between  two 
quantities,  and  2i  geoimetrical  ratio  expresses  the  quo^ 
tient  arising  from  the  division  of  the  two  quantities; 
this  latter  is  called  the  index,  when  referred  to  the 
geometrical  ratio. 

§  79.  The  simplest  reflection  leads  to  the  idea : 
that  two  or  more  such  ratios  may  be  exactly  equal 
to  each  other,  as  well  as  two  quantities  in  general : 
such  an  equality  of  ratio  is  called  a  proportion. 

This  principle  between  two  ratios  is  expressed 
very  naturally  b^v  the  sign  of  equality  between  them, 
as  for  example  : 


*  The  propriety  of  theyp  denominations  is  »ot. worth  discussing  : 
tjijiey  are  mere  names,  ti^  which  the  idra  above  explained  is  to  he 
attached,  wJiich  forms  what  ia  called  their  definition. 


i 


ELEMENTARY  CONSIDfiKATIOMS  OF  RATIO.     Ill 

An  arithmeticai  proporliou  will  be  expressed  thus  : 
7  -  3  =  12-8 
This  says  :  the  difference  between  7  and  3  is  equal  to  the 
difference  between  12  and  8. 

A  geometrical  proportion  will  be  expressed  thus  : 

12  :  3  =  16  :  4  ;    or  »|  =  V 
And  this  says  :  the  quotient  of  12  divided  by  3  is  equal 
to  the  quotient  of  Id  by  4,  as  it  is  evidently  in  both  ra- 
tios  =  4  ;  and  this  is  th;^refore  also  the  index  of  the  two 
equ:d  ratios. 

The  first  term  of  a  ratio  is  called  the  antecedent, 
the  secondXhe  consequent ;  the  fi^  st  and  last  terms  of 
a  j)roportioii  are  called  the  extreme  terms,  the  second 
and  third  the  mean  terms* 

A  neai'er  iinestigation  of  the  properties  of  these 
ratios  will  justify  the  assertion  made  above,  for  we 
sliall  find;  that  the  arithaieticaJ  propoi-tion,  express- 
ed as  a  diffe]*ence,  gives  also  an  equality  of  sums ; 
and  the  equality  of  tiie  quotients  an  equality  of  pro- 
ducts ;  and  that  in  this  property  lies  their  extensive 
utility  in  all  calculations. 

§  80.  It  may  be  easily  seen  that,  while  in  the 
preceding  partof  arithnjetic,  grounded  upon  combi- 
nation only,  we  were  limited  to  things  of  the  same 
kind.  We  obtain  by  this  extension,  or  rhe  considera- 
tion of  the  relation  of  two  things  to  each  other  in  res- 
pect to  quantity n  the  means  of  forming  conclusions  by 
calculation  from  things  of  different  nature  mutually 
acting  upon  each  other;  by  the  condition,  or  simple 
consideration,  of  the  equality  of  the  ratio  of  two  things 
of  one  kind,  to  two  things  of  another  kind,  which  we 
observe  in  nature  in  all  things ;  for  we  may  see  a 
herd  of  cattle,  as  much,  or  as  many  times,  largex* 
than  another  herd  of  cattle,  as  the  money  owned  by 
one  man  is  as  much,  or  as  maviy  times,  larger  than 
the  money  owned  by  anotlier  man ;  a  mountain  as 
much  or  as  many  times  higher  than  a  house,  as  the 


112    ELEMENTARY  CONSIDERATIONS  OF  RATIO. 

amount  of  one  bill  of  exchange  is  of  as  much,  or  as 
many  times,  a  greater  amount  than  anotlier. 

These  considerations  are  daily  made  in  common 
life,  by  every  one,  and  they  need  only  be  transferred 
into  the  language  of  arithmetic,  to  direct  us  in  the 
principles  of  calculation  derived  from  them. 

The  first  of  these  ratios  and  proportions,  namely 
the  arithmetical,,  are  naturally  more  limited  in  their 
application  to  practical  purposes,  as  they  are  the 
result  of  a  more  limited  scale  of  combination.  The 
second,  namely,  the  geometrical,  are  much  more 
extensive,  depending  on  a  higher  scale  of  combina- 
tion; the  geometrical  proportion  is  the  principle  of 
what  is  caJJed  in  arithmetic  the  rule  of  three, 

§81.  I  have  thought  proper  to  enter  into  these 
elementary  deductions,  though  their  aim  is  thereby 
kept  back  for  a  short  time,  because  it  is  all-important 
in  any  study  to  conceive  the  fundamental  ideas  in 
their  generalization,  by  which  the  explanation  is 
so  much  facilitated,  as  ultimately  to  leail  to  a  short- 
ening of  the  task,  both  of  teaching  and  of  studying. 
To  render  these  fundamental  ideas  useful,  we  shall  in 
the  first  place  show  the  consequences  which  lie  in 
them,  from  the  principles  of  combination  upon  which 
they  are  grounded^  and  the  condition  of  equality, 
which  forms  the  ^particular  nature  of  a  proportion. 
We  may  already,  from  the  simple  enunciation  in 
signs,  as  it  appears  above,  conclude  :  that  their  ap- 
j)lication  to  practice  consists  in  the  evident  pro- 
perty, that  any  three  of  the  quantiiies  so  conditioned 
being  given,  the  fourth  is  necessarily  determined; 
the  manner  in  which  tliis  is  done,  thus  rendered 
of  practical  use,  will  appear  from  the  investigation 
of  the  properties  resulting  from  the  principles  of 
proportion. 


AKITHMETJCAI.  FROFOBTIOX.  115 

CHAPTER  II. 

Arit/nnetical  Proportion, 

5  82.  In  Arithmetical  Proportion  the  principle 
evidently  is :  tliat  the  difference  (or,  as  shown 
equally  v  ell,  the  sum)  of  two  quantities  be  equal  to 
the  difference  (or  sum)  of  two  others.  Therefore  if 
each  ratio  is  increased  or  decreased  by  the  same 
quantity,  the  prhiciple  of  equality  continues  to  sub- 
sist as  before,  because  the  quantities  employed,  and 
the  ratios  themselves,  are  both  equal ;  as  it  is  evident 
that  a  proportion  expresses  only  a  quantity  in  the 
form  of  the  difference  (or  the  sum)  of  the  two  others  ; 
thence  we  iiave,  for  instance,  from  the  preceding 
arithmetical  proportion, 

7  —  3  =  12  —  8 
by  adding  on  both  sides  the  number  8, 

74-8  —  3  =  12  —  8-f8 
and  by  again  adding  3  on  both  sides, 

7  ^  S  —  3-1-3  =  12  —  8-f8-fS 
And  as  we  have  seen  that  addition  and  subtraction  are 
inverse  operations,  and  therefore  compensate  each 
other,  the  -f  and  —  also  compensate,  when  they 
are  affixed  to  the  same  quantities ;  therefore  the 
-1-3  —  3  on  one  side,  and  the  4-8  —  8  on  the 
other,  reduce  both  these  numbers  to  nothing,  and  our 
arithmetic  proportion  is  changed  by  it  into 

r  -{-  8  ==  12  4-  3 
an  expression  exactly  of  the  kind  that  it  has  been  said 
(§  78)  could  also  be  used  for  expressing  the  arithmetic 
proportion ;  this  result,  expressed  in  words,  gives  the 
fundamental  property  of  arithmetic  proportions,  that : 
in  any  arithmetical  proportion^  the  sum  of  the  two  ex- 
treme terms  is  equal  to  the  sum  of  the  two  mean  terms. 

If  we  had  expresses!  the  arithmetic  proportion  as 
a  sum,  as  shown  above,  we  would  have  the  result: 
that  the  difference  of  the  extremes  is  equal  to  the  dif- 
10* 


h 


114  ARITHMETICAL   PROPORTION. 

ference  of  the  ineans,  by  the  simple  principle  of  the 
two  arithmetic  operations  of  addition  and  subtrac- 
tion being  opposite  to  each  other* 

From  the  above  result  we  are  authorised  to  con- 
clude :  that  any  operation  of  arithmetic,  performed 
equally  on  both  equal  ratios,  leaves  the  principle  of 
the  ratio  unchanged;  that  is,  equality  will  exist 
between  them  notwithstanding;  and  by  this  princi- 
ple are  guided,  and  of  course  deduced  with  full 
authority,  any  changes  in  the  parts  that  may  become 
necessary, for  a  given  aim,  in  practical  calculation  ; 
thus  it  is  evidently  allowable  to  malte  the  followin.^ 
changes  in  the  above  arithmetical  proportion  : 
As  :      original,  7  —  3  =  12  —  8 

changed  as  above,  7-{-8  =  12-}-^ 

by  adding  equals,  7  —  3-1-6=  12  —  8+6 

subtracting  equals,  7  —  3  —  2  =  12  —  8  —  S 
multiplying  by  equals,  5X7  -3X5  =  12X5-8X5 

7  3  12  8 

dividing  by  equals,  -   —  -    =   —  —  - 

4  4  4  4 

Upon  the  same  principles  the  places  of  the  terms  may 
be  interchanged,  by  tranposing  the  two  extremes, 
or  the  two  means,  or  both,  mutually ;  either  of  the 
proportions  resulting  will  give  the  um  of  the  mean 
terms  equal  to  that  of  the  extremes,  as  is  the  prin 
ciple  in  the  original  proportion ;  thus  is  obtained : 

r  —  12  =  3  —  8 
8  —  3  =  12  —  7 

8  —  12  =  3  —  7 
all  giving  7  -f  8  =  12  -f  3 

Two  or  more  such  proportions  may  also  be  com- 
posed by  the  addition  or  subtraction  of  the  terms, 
respectively  term  for  term ;  thus  the  following  two 
arithmetic  proportions  will  give  results  as  follows : 


{; 


vK  ARITHMETICAL   PROPORTION*.  115 

we  have  from  them 

(7+  16)  — (3  4-4)  =  (12+  19)  — (8  +  7) 
(16  — 7)— -(4 — 3)  =  (19— 12)  — (r  ~  8) 

which  gives  again  the  sum  of  the  extremes  equal  to 
the  sum  of  the  means,  which  is  tlie  fundamental 
principle  of  this  proportion. 

§  83.  In  these  principles  and  comhinations  or 
mutations  lie  the  means,  by  which  numbers  thus 
related  to  each  other,  are  made  susceptible  of  calcu- 
lation ;  their  mutual  dependance  therefore  shows : 
that,  when  any  three  of  them  are  given,  the  fourth  is 
necessarily  determined,  therefore  calculable,  ac- 
cording to  the  principles  here  explained.  The 
arithmetical  process  resulting  from  them  is  evident, 
for  if  from  one  of  the  above  sums  of  extremes  or 
means,  we  subtract  either  of  the  terms  of  the  other 
sum,  we  shall  have  the  result  equal  to  the  other 
term  of  the  latter  sum,  or  what  is  called  the  fourth 
term.  Thus,  if  we  had  in  the  above  original  aritii- 
metical  proportion  the  three  first  terms  given,  as 

.  r'—s  =  12  — 

making  the  sum  of  the  mean  terms, 

3  +  12  =  15 
and  subtracting  from  it  the  first  (or  the  given)  ex- 
treme, namely,  7,  we  have 

15  —  7  =  8 
and  then  the  complete  proportion 

7 — 3  =  12  —  8  as  above. 
§  84.  When,  in  such  an  arithmetical  proportion, 
the  same  number  which  has  been  the  consequent  in  the 
first  ratio,  is  the  antecedent  of  the  second  ratio,  the 
proportion  is  called  a  continued  arithmetical  propor- 
tion; as  in  the  following  : 

12—  10  =  10  — 8 
The  middle  term,  which  is  repeated,  is  called  the 
arithmetical  mean ;  it  is  of  course  equal  to  half  the 
sum  of  the  extremes ;  we  have,  for  instance,  here 


116  GEOMETRICAL   PBOPORTIOK*. 

12  4-  8  =  10 -f  10  =  2  X  10 

12  +  8          20                       2  X  10 
or         = =    10   =  =   10 

2  2  -2 

Such  a  proportion  may  evidently  be  continued 
through  a  whole  series  of  numbers,  as  follows  ; 
12-10=  10 -8=8-6  =  6  -4=4-2  =  2-0 
Then  the  numbers  12;  10;  8;  6;  4;  2;  0;  are 
said  to  be  in  continued  arithmetical  proportion; 
and  the  series,  thus  resulting,  is  called  an  arithme^ 
ileal  progressioih  or  an  arithmetical  series.  Their 
use  is  very  frequent  in  higher  calculations,  and  we 
shall  treat  of  them  hereafter;  we  will  here  only 
state,  that  the  successive  numbers  may  either  iri- 
crease  or  decrease  according  to  the  same  principle^ 
and  that,  from  the  nature  of  their  application  in 
practice,  they  are  always  written  in  the  manner  wc 
have  stated  that  arithmetical  proportion  might  be 
written;  namely,  with  the  sign  of  addition  :  thus 

12  -f  ^0  4-8  +  6 +4 +  2 +  0 
would  be  a  decreasing  arithmetical  progression^  or 
series;  and 

3  +  5  +  7  +  9  +  11  +  13  +  15  +  17 
an  increasing  arithmetical  series,  or  progression; 
both  are  subject  to  the  same  laws,  and  the  same 
principles  for  the  mutual  detei-mination  of  their 
ajeveral  parts  from  each  other,  as  we  shall  seeinits 
proper  place, 

CHAPTER  III. 

Geometrical  Proportion, 

<5  85,  The  principles  of  Geometric  Ratio,  as  we  have 
seen  abovC;,  take  their  rise  in  the  combinations  of 


I 


GEOMETRICAIi  PROPORTIOIf.  117" 

the  second  kind,  explained  in  Part  I.,  that  is,  from 
multiplication  or  division.  In  it  therefore  the  ratie 
is  convsidered  as  the  indication  of  how  many  times 
a  quantity  is  greater  or  smaller  than  anotlier ;  the 
quantity  indicating  this  ratio  in  one  single  number 
is  called  the  index  of  the  ratio ;  it  is  exactly  the 
same  as  the  quotient  in  a  fraction  or  in  a  divisiati. 

The  investigation  of  the  consequences  of  this 
principle  in  a  geometrical  proportion  gives  the  gene- 
ral law,  which  must  guide  all  the  operations  founded 
upon  geometrical  proportion,  and  lead  to  the  disco- 
very of  all  its  properties.  For  this  purpose  it  is 
best  to  present  the  geometric  proportion  as  an 
equality  of  fractions^  or  quotients^  which  we  have 
found  it  to  be  ;  thus  we  have 

12  :  3  =  16  :  4 
12  16 

or  —  =  — ;  evidently  presenting 

3  4 

the  identity  4=4      by*  the  execution  of 

the  division,  and  indicating  4  as  the  index  or  the. 
quotient. 

Reducing  the  two  fractions  to  a  common  denomi- 
nator, we  obtain,  without  any  change  in  the  value, 
(as  proved  in  fractions) 

12  X  4  16  X  3 


3X4  3X4  1/ 

On  account  of  the  whole  fractions  or  quotients 
being  equal,  from  the  nature  of  geometric  propor- 
tions, and  at  the  same  time  also  the  denominators  of 
the  fractions  obtained,  it  is  a  necessary  consequence, 
that  the  numerators  must  also  be  equal. 

Therefore  12X4  =  16X3 

which  is  evidently  identical  with  48  =  48 

Comparing  this  result  with  the  geometrical  pro- 
portion given,  we  obtain  the  proof  of  the  essential 


lis  GEOMETFICAL  PBOPOKTIOIf. 

proper  ttj  of  geometrical  proportions;  tliat  the  jiroducl 
of  the  txvo  extreme  terms  is  equal  to  the  product  of  the 
two  mean  terms.  A  property  exactly  analogous  to 
that  obtained  for  tlte  arithmetical  proportion,  which 
in  that  case  relates  to  the  sum  of  the  terms,  and  in 
this  to  the  i)roduct. 

This  at  the  same  time  confirms  the  general  princi- 
ple stated  above  :  that  a  geometric  proportion  might 
be  equally  well  expressed  by  a  product,  as  by  a 
quotient,  and  by  operations  tlie  converse  of  those 
made  above,  it  would  lead  to  the  expression  of  an 
equality,  by  division,  quotient,  or  what  is  usually 
called  ratio  We  would  in  that  way  of  representing 
the  proportion  obtain,  by  dividing  both  sides  suc- 
cessively by  3  and  4,  obtain  : 

12  X  4  16  X  3 


3X4  S  X  4 

And  because  the  4  in  the  one  fraction,  and  the  3  in 

the  other  fraction,  compensate,  by  division  in  the 

numerator  and  denominator,  we  have  from  this : 

12         16 

3  4 

or  12  :  3   =    I6  :  4 

that  is,  the  identical  expression  of  the  usual  geome- 
trical proportion.*-  - 

-■^^ ' 

*  The  iiiaiheiicati*  ii  ('Xjiression  ol  these  two  modes  of  presenting 
the  geometrical  proportion  would  be  ;  by  the  products:  thai  the 
jnoportion  is  an  tqualiltj  of  products ;  and  by  the  usual  mode  it  is : 
a  proportion  is  an  equalily  of  quoiitnis ;  this  cannot  escape  the 
notice  of  any  one  reflertin«;  upon  the  prirrciple  stated  at  the  very 
cmtset :  that  in  all  arithmetical  principles  of  operatioD,,the  system 
must  be  g^ood,  or  hold  true,  both  directly  and  conversely.  The 
sign  of  equality  between  the  two  ratios  forming  a  proportion,  is 
therefore  the  only  proper  siajn,  and  the  four  dots  used  by  many 
authors  are  against  principles,  because  they  do  not  convey  the 
idea  of  the  principle,  a  thing  so  esseatial  to  actual  knowledge 
So- to  write,  for  instance,  12  :  3  :  :  16  :  4  is  wrong,  or  at  least  a 
pleonasmus  of  signs,  leading  into  misapprehensions,  a  thing  coa> 
trary  to  principles  ia  exact  science. 


i 


GEOMETRICAL    PROPORTION.  119 

•  §86.  The  principle  now  deduced,  and  proved, 
givfes  all  the  consequences,  which  are  so  useful  in 
the  application  of  proportions  to  practical  calcula- 
tion ;  namely  :  that  in  a  geometrical  proportion,  all 
those  mutations  are  admissible^  which  do  not  alter  the 
principle,  that  the  product  of  the  two  extreme  terms 
is  equal  to  the  product  of  the  two  mean  terms, 
♦  Therefore  we  can  make  all  the  changes  shown 
above,  in  relation  to  the  example  before  used. 

From     12:3  =  16:4 

1  ,    rp  ,.     ^middle     },  U2:16=3:4 

1st.  Transposing  the  ^  ^^^^^^^  ^  terms   ^  ^.3  ^  ,^.  ^^ 

2d.  Changing  antecedents  into  consequents  3:12=  4:  IG 

These  are  all  evident,  from  the  simple  principle ; 
that  the  products  of  two  quantities  are  the  same, 
whichever  of  the  two  be  the  multiplier,  or  multipli- 
cand ;  that  is,  because  3  times  4  is  the  same  as  4 
times  3^  as  well  known ;  or  any  two  other  numbers  ^ 
they  all  equally  present:  3  X  16  =  4  X  12  =  48. 

3d.  Multiplying  by  the  same  number  either 
both  antecedents,  as        2X12:3  =  2x16:4 
or  both  consequents,  as       12:2X3=16:2X4 
or  all  the  terms,  as        2X12:2X3=:2xl6:2x4 

The  results  must  evidently  preserve  the  principle 
of  equality  of  products  of  extremes  and  means;  be- 
cause in  every  case  the  same  multiplier  is  contained 
in  each  product ;  for,  though  tlic  first  product  appa- 
rently presents  other  numbers,  the  identity  of  their 
'•osult  reduces  them  to  the  same  principle. 

4th.    Dividing  in  the  same  manner  as  before  will  give 
.)  the  same  order  ; 

V^  :  3  =  V  :  4 

12:1  =  16:1 

12*3    .  n    •    4 

¥     •    2    3      •    2 

To  which  the  same  reasoning  applies  as  to  the  multi- 


ISO  GEOMETRICiX  PROPOIlTIO]!f. 

plication ;  and  it  is  proper  to  make  this  division  in  all 
cases,  where  the  data  of  a  proportion  are  compounded 
of  numbers  having  common  measures,  in  the  terms 
forming  the  numerator  and  the  denominator  of  the 
final  result. 

We  can  also  compose  and  decompose  the  geome- 
trical proportion  by  its  antecedent  and  consequent 
terms,  in  such  a  manner  as  to  obtain  the  proportion 
between  their  sum  or  difference  with  the  antecedents 
or  consequents,  or  between  these  sums  and  differ- 
ences themselves,  which  furnishes  an  additional 
means  of  calculation  for  a  number  of  practical  cases. 

6th.  Thus  we  obtain  from  our  example  the  following 
results  of  mutations  ;  viz  :  ^ 

By  adding  the  antecedent  and  consequent  and  compar- 
ing them  with  the  antecedents  : 

12  +  3  :  12  =  16  -f  4  :  16 
By  comparing  the  same  with  the  consequents  : 

12  4-  3  :  3  =  16  +  4:  4 
By  comparing  the  differences  of  the  antecedents  and 
the  consequents  with  the  antecedents  : 

12  — 3  :  12  =  16  -  4  :  16 
By  comparing  the  same  with  the  consequents  : 

12  -  3:3=  16  -4  :4 
By  comparing  these  sums  and  differences  themselves  : 
12  +  3  :  12  -  3  =  16  4-  4  :  16  -  4 

All  these  compound  proportions  have  necessarily 
the  property  of  giving  equal  products  of  the  ex- 
treme and  the  mean  terms,  because  they  always- 
contain  only  a  different  combination  of  the  factors, 
giving  equal  products,  exactly  as  in  the  simple 
proportions 

All  the  mutations  under  Sd,  4th  and  5th,  again 
admit  of  course  the  exchange  of  the  places  of  the 
extreme  and  the  mean  terms,  which  the  original 
proportion  admits.  Any  one  of  these  mutations  is 
to  be  applied  either  to  disengage  one  of  the  quanti- 


GEOMETRICAL   PBOPOKTION.  121 

ties  contained  in  a  given  proportion,  or  whenever  it 
can  lead  to  an  abridgment  of  the  statement ;  and  it 
will  be  found  that  in  proportions  apparently  com- 
pound they  often  lead  to  the  final  solution,  without 
its  being  necessary  to  have  recourse  to  both  the 
multiplication  and  division  of  the  terms  themselves, 
only  the  one  or  the  other  of  the  operations  remain- 
ing to  be  performed ;  that  is,  the  one  or  the  other 
term  is  reducible  by  it  to  unity  ;  the  future  applica- 
tion will  show  their  use  by  examples  in  given  cases. 
5  87.  If  we  have  two  geometrical  proportions, 
they  may  be  multiplied  together,  or  divided  the  one 
by  the  other,  term  by  term,  with  equal  correctness  of 
conclusion ;  for  it  is  the  same  as  multiplying  two 
equal  fractions  by  two  other  equal  fractions,  the  pro- 
ducts of  which  will  again  be  equal ;  therefore,  ac- 
cording to  the  principles  first  deduced,  the  products 
of  the  extreme  and  mean  terms  will  again  be  equal. 

For  example,  let  the  two  followiDg  proportions  be 
thus  composed  ;  viz  : 

18  :  6  =  12  :  4  or  the  fractions   Y  =  ¥ 

and    15  :  3  =  25:  5  "         "  V  =  ¥ 

Multiplying   the   proportions  term  by  term,    or   equal 
fractions  by  equal  fractions,  we  obtain  : 

18X15        12X25 

18X16:6x3  =  12  X  25  :  4X5;  or— ==  . 

6x3  4X5 

where  the  product  of  extremes  and  means  gives 
18X15x4x5  =  6X3x12X25 
5400  =  5400 
and  by  reducing  the  fractions,  by  means  of  their  common 
measures  :  15  =   15 

In  like  manner,  by  division,  we  would  obtain  from  the 
foregoing 

18     6        12     4  18  X  3        12  X  5 

16     3        25     5  6X15       4  x  26* 

11 


122  GEOMETRIC  AX   PROPORTION. 

giving,  by  products  of  extremes  and  means, 
6  X  12        4  X  18        24        24 


3  X  26        6x15        26        25 
or  as  fractions,  f  =  !• 

all  equally  leading  to  identical  results. 

Supposing,  therefore,  six  terms  in  these  two  pro- 
portions given,  in  any  manner,  the  t\^o  remaining 
terms  may  be  determined  from  them.  And  in  ge- 
neral :  as  many  proportions  as  are  given,  so  many 
unknown  quantities  may  be  determined  by  them. 

This  is  is  also  the  principle  of  what  is  called  in 
arithmetic  the  Compound  Rule  of  Three,  It  may 
be  carried  to  any  length,  by  further  combination 
upon  the  same  principles ;  when  it  is  carried  through 
a  number  of  proportions,  to  determine  only  one  un- 
known quantity,  it  is  called  the  chain  rule.  The 
application  of  both,  and  their  extensive  utility,  will 
be  shown  in  their  proper  places. 

The  proportion  may  also  be  multiplied  into  itself 
term  by  term ;  and  thereby  may  be  obtained,  from 
the  proportions  of  lineal  dimensions,  the  proportion 
of  the  superficial  dimensions  corresponding  to  them, 
or  the  squares.  By  the  products  of  three  such  equal 
proportions  term  by  term,  will  be  obtained  the  pro- 
portion of  the  solids  having  the  same  lineal  dimen- 
sions for  their  sides,  or  the  cubes.  Thus  would, 
for  instance,  be  obtained  : 

From  the  simple  proportion    18:6  =  12:4 
the  square,  18 X  18  :  6X6  =  12X12  :  4X4 

or  328  :36  =  144  :  16 

the  cubes, 

18x18x18:6x6X6  =  12x12x12:4X4X4 
dr  5904  :  216  =  1728:64 

§  88.  It  may  readily  be  conceived  :  that  in  geo- 
metrical proportions  a  continuance  may  take  place,. 


GEOMETHICAI*  PBOPOKTIOW.  1^3 

Hs  well  as  in  the  arithmetical ;  that  condition  may  be 
again  expressed  by  the  equality  of  the  two  middle 
terms ;  as  follows  : 

16:8  =  8:4;  which  gives   8  X  8  =  16  X  4 
as  the  products  of  extremes  and  means.    The  middle 
term  is  called  the  geometrical  mean. 

To  this  every  property  applies  that  belongs  to 
general  proportion ;  it  therefore  admits  all  the 
changes  heretofore  shown.  The  product  of  the  two 
mean  terms,  being  compounded  of  two  equal  factors, 
presents  what  is  called  a  square  number;  comparing 
it  by  this  to  the  rectangular  surface  which  would 
have  all  its  sides  equal,  and  showing  the  reduction 
of  a  rectangular  figure  of  two  unequal  sides  into  a 
square. 

5  89.  Such  a  proportion  may  evidently  be  conti- 
nued by  either  increasing  or  decreasing  numbers 
as  well  as  an  arithmetical  one ;  producing  quanti- 
ties having  a  common  factor,  which  is  called  the 
common  index^  or  constant  ratio ;  and  the  progres- 
sion or  series  resulting  from  it  is  called  a  geometric 
eal  progression  or  series.  In  the  increasing  progres- 
sion the  common  index  is  a  whole  number,  and  in 
the  decreasing  one  It  i»  trrid^ntly  a  frartimi  •  it  cop- 
responds  likewise,  as  in  the  ratio  itself,  to  the  quo- 
tient arising  from  the  division  of  two  successive 
terms. 

The  following  is  an  example  of  such  a  progres- 
sion or  series : 

64  :  32  =  32  :  16  =  16  :  8  =  8  :  4  =  4  :  2  =  2 :  1  = 
1-i  =  l'l  —  1.1  —  1.     ^f 

2  2*4  4*"8  f'TF 

This  is  also  usually  written  omitting  the  signs  of 
equality,  and  the  terms  are  separated  by  the  sign  of 
addition  (-f )  instead  of  the  sign  of  division  ( : ),  be- 
cause  this  notation  is  better  adapted  to  the  use  made 
of  these  series  in  higher  calculations,  where  they 
are  of  great  utility ;  the  above  series  may  then  be 
written  thus : 


^24  EVLE    OF  THREE. 

5  =  64 -f- 32  +  16  +  8 +4  +  2  +  1 -f  i+f +|4-tV+ ^G- 
Every  subsequent  number  being  here  the  half  of  the 
preceding  one,  the  common  index  ot  the  series 
is  =  ^  ;  or  any  one  of  the  numbers  multiplied  by  I 
will  produce  the  number  immediately  succeeding  it. 
It  is  proper  here  to  drop  this  subject  for  the  pre- 
sent in  order  to  take  it  up  in  a  later  part  of  the  work, 
when  we  shall  investigate  its  consequences  and 
practical  applications. 


CHAPTER  IV. 

Rule  of  Three, 

§  90.  In  the  preceding  chapter  we  have  found ; 
that  geometrical  proportion  is  the  same  with  the 
equality  of  two  fractions,  and  that  the  products  of 
its  extreme  and  mean  terms  are  equal.  We  proceeded 
in  the  demonstration  thus :  the  numerator  and  deno- 
minator of  the  two  equal  fractions  were  multiplied 
each  by  the  denominator  of  the  other,  equal  denomina- 
tors being  obtained  by  it,  the  conclusion  was  thai 
tlie  numerators  arp.  also  equal. 

If,  instead  of  multiplying  both  factors  by  the  de- 
nominators mutually,  we  multiply  only  one  in  nume- 
rator and  denominator,  the  equality  will  evidently 
remain,  because  the  value  of  the  fraction  so  multi- 
plied does  not  change.  Thus  we  obtain  from  the 
proportion 

12  :  3  =  16  :  4  or,  expressed  as  a  fraction,  y  =  V* 
by  multiplying  the  first  fraction,  in  numerator  and 
denominator,  by  the  denominator  of  the  second, 
12  X  4  16 


3X4  4 

and  by  operating  equally  upon  the  second  fraction* 


mXTLE  «*  THBBE*  125 

12  3  X  16 


i 


3  3X4 

In  botli  cases  the  two  fractions  having  one  of  the 
factors  in  the  denominator  equal,  the  same  prin- 
ciple applies  to  this  equal  factor,  as  to  the  equal 
denominators,  according  to  what  is  known  of  the 
principles  of  fractions ;  they  therefore  compensate 
each  other  in  this  equality,  and  we  obtain 
12  X  4 

y  the  first:  ==  16 

3 

16  X  3 
and  by  the  second :    12  =  _— — 

4 
That  is  :  we  obtain  one  of  the  terms  expressed  by 
the  three  others ;  and  this  in  such  a  manner,  that  the 
product  of  either  ecctremes  or  means  being  made, 
and  this  divided  by  the  one  mean  or  extreme,  the 
result  gives  the  other  mean  or  extreme. 

As  we  have  seen :  that  the  mutations  allowed  in 
geometrical  proportion  admit  any  one  term  to 
be  made  either  extreme  or  either  mean,  under  the 
corresponding  mutations  of  the  other  terms,  we  can 
generally,  by  dividing  any  one  of  the  products  by 
one  factor  of  the  other,  obtain  a  result  equal  to  the 
other  factor  of  that  product. 

Thus  we  would  deduce  from  the  above  proportion 
all  the  following  results ;  viz : 
12x4  16X3  12x4  16X3 

3  '4  '       16  '12 

This  is  the  complete  principle  and  mode  of  per^ 
forming,  what  is  called,  the  rule  of  three,  from  the 
circumstance  that  three  quantities,  or  numbers,  are 
used  to  determine  a  fourth. 
If  therefore  any  ratio  between  two  known  qiian*^ 
11=^ 


126  RCXE    OF   THREE. 

titles  is  said  to  be  the  same  as  (or  equal  to)  the  ratio 
between  one  other  known  quantity  and  an  unknown 
one,  the  above  principle  gives  the  determination  of 
this  unknown  quantity  by  the  above  process,  adapt- 
ed to  the  given  case  or  question,  and  any  of  the 
mutations  shown  in  the  preceding  chapter  can  be 
applied  to  it,  as  may  be  required. 

§  91.  We  will  now,  authorised  by  the  foregoing 
proofs,  make  the  application  of  the  principles  of 
geometric  proportion  to  the  practical  irperations  of  the 
rule  of  three.  As  it  will  often  be  necessary  to  act 
upon  the  unknown  quantity  as  if  we  knew  it,  in 
order  to  make  such  of  the  above  demonstrated  muta- 
tions as  may  be  required,  we  shall  here  introduce 
the  method  so  advantageously  practised  in  universal 
arithmetic,  namely,  to  denote  the  unknown  quantity 
by  a  letter,  and  choose  for  that  always  one  of  the 
last  letters  of  the  alphabet,  as  x,  y,  &c. ;  and  when 
we  shall  have  this  letter  alone  on  one  side  of  the 
sign  of  equality,  we  have  seen  from  what  has  already 
been  said,  that  the  unknown  quantity  is  determined 
by  the  combinations  of  the  known  ones  presented  on 
the  other  side  of  this  sign  of  equality ;  that  is,  the 
number  obtained  by  them  will  be  the  value  of  this 
unknown  quantity ;  this  is  but  a  small  extension  of 
the  use  of  signs  to  denote  the  operations  of  arithme- 
tic, which  has  been  introduced  in  the  very  begin- 
ning, and  found  so  useful  in  expressing  distinctly 
the  operations  of  arithmetic. 

Though  it  is  evidently  indifferent  in  which  of  the 
four  places  of  the  proportion  the  unknown  quantity 
stands,  a  habit  prevails,  of  stating  the  proportion  so 
that  the  unknown  term  occupies  the  fourth  place  ia 
the  proportion ;  we  shall  follow  it,  wherever  the 
combinations  do  not  present  reasons  for  another  ar- 
rangement. 

Isi  Example.  To  determine  the  unknown  quantity  in 
the  proportion  1 6  :  7  ==  ( 9  :  a- 


r 


RULE    OF   THREK,  127 

The  product  of  the   two   mean  terms  divided  by  the 
tirst  extreme  will,  as  proved  above,  give  the  value  of  x, 
or  the  other  extreme,  which  is  the  quantity  sought ;  thus 
7  X  19  133  13 

=a;  =  --=8H =  8,  8666  +  &c. 

15  15  16 

which,  placed  in  common  examples,  as  has  been  fully 
ahown  in  multiplication  and  division,  stands  thus  : 
19 
7 

133  13 

=  8  -f  ~ 

15  15 

13 
or  by  continuing  the  division  into  decimal  fractions  : 
133 

=  8,8666  -f  &c. 

16 
130 
100 
100 
10 
when  the  division  continued  would  evidently  give  a  con- 
tinued succession  of  the  6. 

Thus  therefore,  the  fourth  term,  or  x,  is  deter- 
mined ;  and  any  other  proportion,  or  rule  of  three, 
the  terms  of  which  are  ever  so  great  or  complicated, 
may  be  solved  by  the  same  operations,  performed 
upon  the  respective  numbers. 

2d  Example.  Suppose  that  7  men  mow  37  acres  of 
meadow  in  a  certain  time ;  how  many  acres  will  27  men 
mow  in  the  same  time  ? 

Here  we  have  given  :  the  ratio  between  the  tneji  em- 

ployed,  to  which,  by  the  nature  of  the  subject,  the  ratio 

between  the  acres  of  meadow,  mowed  by  each  number  of 

men  respectively,  must  be  equal ;  of  this  only  the  number 

if  acres  mowed  by  the  7  men  is  given,  and  the  number 

I"  acres  that  can  be  mowed  by  27  men  is  the  quantity 


i28  KUIE   OF  THREE. 

sought,  which  we  have  agreed  to  designate  first  by  a 
letter,  as  x. 

If  therefore  we  make  the  number  of  men  correspond- 
ing to  the  number  of  acres  given,  the  first  antecedent  term 
of  the  geometric  proportion,  the  second  number  of  men 
will  be  the  first  consequent,  or  second  term  of  the  pro> 
portion  ;  the  antecedent  of  the  second  ratio,  that  is,  that 
of  the  number  of  acres  mowed  in  each  case,  must  be  the 
37  acres  ;  as  corresponding  to  the  work  of  the  number  of 
men  forming  the  antecedent  in  the  first  ratio  ;  the  num- 
ber of  acres  corresponding  to  the  number  of  men,  whose 
work  it  is  intended  to  ascertain  by  the  operation,  here 
our  re,  must  therefore  be  the  consequent  of  the  second 
ratio,  or  the  fourth  term  of  our  proportion.  This  gives 
therefore  the  statement : 

Men.   Men.       Acres.  Acres. 
1  :  21    =    37   :  x 
And  by  the  operation  shown  above,  and  deduced  before^ 
we  obtain  : 

Acres.  Acres. 

21X31  6 

X  =  =    142  -}-  -  =   142,714  &c. 

7  7 

where   the  decimal  fractions   are  evidently  carried  far 
enough  for  any  practical  purpose  in  the  case. 

I  have  been  thus  long  and  detailed  in  this  first  exam- 
ple of  the  application  of  geometric  proportion  to  the  rule 
of  three,  to  show  the  details  of  the  reasoning  which  must 
guide  in  the  statement  of  a  practical  question  ;  that  I 
may  be  allowed  in  future  to  suppose  them  known,  and 
that  1  may  have  to  explain  only  the  peculiarities  which 
may  occur  in  other  cases,  in  the  same  manner  as  I  here 
suppose  the  arithmetical  operations  of  multiplication  and 
division  as  sufficiently  explained  in  the  first  example. 

The  scholar  will  now  observe  :  that  in  performing 
the  arithmetical  operations,  the  things  or  objects, 
which  the  numbers  represent,  do  not  enter  into  the 
consideration,  and  that  the  numbers  alone  are  treated, 
as  indicative  of  the  relation  of  these  things  in  regard 
to  quantity,  according  to  our  first  definition  of  quanti- 


I 


BVXE    OF   THREE.  129 

ty;  for,  what  would  a  product « *f  men  into  acres  of  land 
represent  in  nature  ?  But  the-  divisif»n  made  again  by 
a  number  rep-  esenting  men,  may  be  considered  as 
compensating,  in  a  maimer  similar  to  that  ot  the  equal 
factors  in  the  numerator*  and  denominator  in  a  frac- 
tion, which  compensate  each  other ;  and  there  then 
remains,  we  might  say,  t!»e  denomination  of  acres  in 
the  numerator,  to  give  tlie  denomination  to  the  result. 
This  is  exactly  analogous  to  what  has  been  said 
at  the  beginning  of  this  part  of  aritlimetic ;  that  the 
ratio  only  of  the  two  things  of  the  same  kind  is 
taken,  as  the  principle  that  determines  the  ratio  of 
two  other  things,  which  may  be  of  a  nature  com- 
pletely different  from  the  two  first.  We  shall  in 
general  find,  in  all  results  of  calculations  relating 
to  objects  of  different  kinds,  that  the  denomination 
of  the  result  is  that  of  the  kind  of  quantity  or  things 
which  appear  in  it  in  an  odd  number  of  terms,  and 
that  those  which  appear  in  an  even  number  of  terms 
act  as  mere  numbers,  giving  no  denomination  to  the 
quantity  of  the  result.  This  remark,  which  is  here 
very  simple,  becomes  of  greater  importance  in  higli- 
er  calculations,  and  is  in  all  cases  an  indispensable 
property  of  an  accurate  result. 

3d  Example.  My  neighbour  bought  372,  45  acres  of 
land  for  ^720,6,  but  I  can  dispotse  of  only  §215,  5  for  that 
purpose  ;  how  much  land  can  I  purchase  at  the  same 
rate  ? 

The  ratio  of  the  money  is  here  given,  and  the  ratio  of 
the  land  purchased  by  it  must  of  course  be  the  same  ; 
we  have  therefrom  the  statement : 

§720,  5  :  §215,  5  =  372,  45  acres  :  x  acres. 

This  proportion  can  be  reduced  to  simpler  numbers  by 

dividing  corresponding  terms  by  5,  which  is  a  common 

factor ;  it  is  therefore  proper  to  do  it ;  thus  it  becomes: 

144,  1  :  43,  I  =  372,45  :  x 

43,  1  X  372,  45 

tvhich  gives    x  = =  111,422  acres. 


ISO  HULE  OF  THREE. 

Here  it  is  evidently  most  proper  to  proceed  altogether 
by  decimal  fractions,  in  which  also  the  answer  fits  best. 

4th  Example.  If  57  lb.  7  oz.  of  spices  be  bought  for 
^17,  25,  what  must  I  pay  for  «7  lb.  16  oz.  7  dwt.  ? 

Here  the  ratio  of  the  spices  is  given,  and  the  quantities 
contain  denominate  fractions ;  we  would  have  to  divide 
the  second  by  the  first,  which  is,  as  shown  above,  a  very 
inconvenient  operation  ;  we  may  therefore  either  reduce 
the  \yeights  to  the  lowest  denomination  of  the  denominate 
fractions,  which  is  the  pennyweight,  and  then  proceed  as 
in  whole  numbers,  or  reduce  thej^e  denominate  fractions 
to  decimal  fractions  of  the  pounds  We  have  seen  above 
that  the  first  is  the  most  convenient,  when  we  do  not 
foresee  that  the  denominate  fractions  will  ^ive  short  and 
determinate  decimals  ;  we  shall  therefore  proceed  by 
this  reduction ;  thus  we  obtain  for  the  two  first  num- 
bers, 57  and  87 
12  12 


114 

174 

577 

87 



10 

691  oz. 

20 

1054  oz. 

2U 

13820  dwt. 

21080 

7 

21087  dwt. 
by  multiplying  first  the  pounds  by  12,  to  reduce  them  to 
ounces,  then  adding  the  ounces  given,  then  multiplying 
by  20,  to  reduce  to  pennyweights,  and  adding  the  penny- 
weights given  ;     thus  we  obtam  the  statement : 

13820  :  21087  =  17,25  :  x 
Dividing  by  5,  to  reduce  : 

2764  :  21087  =  3,  45  :  a; 

3,45X21087 
which  gives  :     x  s=  __— _  =  J26,  32 
2764 


or 

6 

Divid 
which 

ing  by  5, 
i  gives 

1 

and 

x-^ 

and  the  whole 

sum 

SUXE   OF  THBEE.  131 

icimal  fractions  resulting  we  slop  at  the  32 
cents,  no  mills  coming  after ;  further  accuracy  would  be 
useless. 

6th  Example.    A  sum  of  money  being  shared  between 
John  and  James  in  the  proportion  of  9  to  4,  it  results 
that  John   has  §15   more   than  James;  what  were  the 
shares  of  each  ?  and  what  was  the  whole  sum  shared  ? 
The  proportion  stated  from  the  above  data  stands  thus  : 
JohrCs.     James*. 
9  :  4  =  X  -{-  \o  :  X 
Subtracting  the  consequents  from  the  antecedents,  and 
comparing  with  the  consequents,  we  obtain  : 
9  —  4:4  =  i:-f-  15  —  xra; 
:  4  =  15  :  a; 
:  4  =  3  :  a; 
X  =  12  =  James'  share  ; 
15  =  27  =  John's  share  ; 
=  39       has  been  shared. 
6th  Example.     Two   merchants  make  a  joint  stock  : 
they  contribute  in  the  proportion  of  14  to  5;  the  differ- 
ence  between  the  full  shares  is  §504  ;  what  was  each 
individual's  share,  and  the  whole  stock  ? 

( Stock  2064 

which  is  obtained  by  exactly  the  same  process  as  above. 

7th  Example.  Three  merchants  make  a  joint  stock  ; 
the  first  puts  in  a  certain  unknown  part  of  the  capital, 
the  second  2000  dollars  more,  and  the  third  3000  dollars 
less,  than  the  first ;  the  ratio  of  the  shares  of  the  second 
and  third  is  as  9  :  to  5  ;  what  are  all  the  individual 
shares,  and  the  stock  itself? 

If  we  call  the  share  of  the  first,  which  regulates  the 
whole  question,  x,  we  shall  have  the  statement  thus  : 

9  :  5  =  X  -f  2000  :  x  -  3000 
Comparing  the  difference  between  the  antecedents  and 
consequents  with  the  same  consequents,  we  obtain  : 

9  -  5  :  5  =  a;  -f  2000  -  a;  +  3000  :  x  ~  3000 
er  4:5  =  5000  :  x  -  3000 


132  aULE    OF   THREE. 

Dividing  the  antecedents  b}-  4  ;  1  :  5  =s  1250  :  x  —  3000 
whence  5  X  1250  =  a;  -  3000 

6250  =  X  —  3000 
That  is,  §6250  is  the  share  of  the  third. 

The  share  of  the  first  is  therefore  =  g  9250. 

"  "  second         "  =      11250. 

And  the  whole  stock  =     26750. 

8th  Example.  A  bankrupt  leaves  clear  property 
^84421,  26  ;  his  creditors  are  as  follows  ;  viz  : 

Jones  for  $  5629 

Williams  14207 

Rufus  592 

King  29768 

Eldridge  120352 

What  dividend  in  the  hundred,  or  proportional  part,  can 
be  paid,  (under  the  supposition  of  equal  concourse,)  and 
what  will  each  creditor  get  for  his  share  ? 

Here  the  ratio  of  the  sum  of  the  debts  to  the  clear  pro- 
perty will  be  the  constant  ratio,  »which  will  give  the  rule 
for  the  division ;  each  claim  forms  the  second  antece- 
dent, or  what  is  the  same  thing,  the  first  term  of  the 
second  ratio.  Or,  the  fraction  arising  from  the  division 
of  the  property  by  the  sum  of  the  debts,  which  may  be 
most  easily  expressed  in  decimals,  will  be  a  constant 
multiplier  for  each  of  the  individual  debts,  and  the  shares 
will  be  the  product  of  this  fraction  by  the  amount  of  the 
84421,26         42210,63 

claim.      Thus = =  0,495    will 

170548  85274 

be  a  constant  multiplier  for  each  of  the  claims,  which  will 
give  the  shares  as  follows  :         Of  Jones,        §2786,  355 

Williams,     7032, 466 
Rufus,  293, 04 

King,  14735,  16 

Eldridge,   59574,24 

§  92.  In  many  cases  in  nature,  and  the  common 
intercourse  of  life,  the  things  whose  ratio  is  com- 
pared, augment,  the  one  in  the  same  ratio  as  the 


J 


RtriiE    OF   THRBE.  133 

ther  diminishes,  and  inversely;  as  for  instance, 
the  more  men  are  about  a  work,  the  less  time  it 
will  require  to  do  it ;  the  quicker  a  man  walks,  in 
proportion  to  another  man,  the  less  time  he  will 
require  to  go  through  a  certain  space ;  and  so  in 
many  other  cases  in  nature.  That  is  to  say :  the 
ratios  (of  these  things,  or  the  results)  are  inversed. 
Tiierefore,  in  all  such  cases,  the  ratio  of  the  two 
given  terms  of  the  same  kind  is  also  to  be  inverted 
in  the  statement  of  the  proportion,  and  then  the  ope- 
ration of  the  rule  of  three  is  to  be  executed  with 
this  inverted  ratio,  in  the  same  manner  as  above 
with  the  direct  one;  this  operation  is  evidently 
grounded  on  the  nature  of  the  things,  or  the  ques- 
tion ;  as  in  the  following  examples. 

1st  Example.  I  have  a  meadow,  which  6  men  usually 
mowed  in  17  days;  but,  the  season  being  precarious,  I 
wish  to  have  it  mowed  in  3  days  ;  how  many  men  must 
I  employ  ? 

Evidently  the  shorter  the  time,  the  more  men  I  must 
employ,  ^o  the  ratio  of  the  men  is  the  inverse  of  that  of 
the  time  ;  and  as  this  latter  ratio  is  given,  I  must  write  it 
inversely  ;  thus  the  statement  becomes  : 
Days.  Men. 

3  i  \1   =  G   :  X 
or  1   :    17  =  2  :  a; 

giving  a:  =  2  X  17  =  34  men  ; 

and  so  many  men  must  be  employed  to  do  in  3  daj'S  the 
work  of  6  men  in  17  days. 

2d  Example.  Two  men,  starting  at  the  same  time,  ride 
a  certain  distance  ;  John  travels  at  the  rate  of  6^  miles  an 
hoar,  and  Peter  7|  an  hour  ;  Peter  arrives  after  20  hours 
20  minutes  ;  when  will  John  arrive  ? 

The  ratio  of  the  time  of  arrival  is  evidently  the  inverse 
of  that  of  the  speed,  or  number  of  miles  made  per  hour  ; 
therefore  the  statement  must  also  be  inverted  ;  thus : 
Miles.  H.  Min. 


61  :  7f  =  20.  20  :  x 


12 


134  RUXE   OF  THREE. 

Fractions  occurring  here,  they  must  be  reduced  ;  but  20 
minutes  being  a  third  of  an  hour,  and  the  fraction  -^  oc- 
curring in  the  first  term,  we  may  take  advantage  of  it  to 
shorten  this  operation  thus  :  reducing  the  whole  num- 
bers to  fractions  upon  this  consideration,  we  obtain  : 


Multiplying  by  3        19  :  V   =  61  :  ^' 

The  fraction  of  the  second  term  may  be  left  unreduced. 

and  the  result  written  thus  : 

31  X  61         1891 

X  = =  =  24,  88157  hours. 

4X19  76 

As  60  minutes  make  one  hour,  every  tenth  of  an  hour  is 
6  minutes  ;  the  decimals  of  hours  are  therefore  reduced 
to  minutes  by  multiplying  by  6,  and  remarking  that  the 
result  of  the  tenths  gives  the  units  of  the  minutes,  or  the 
denominate  fraction  of  60  parts,  or  ^^^,  the  above  becomes 
thereby  24  h.  52,8492m.  The  same  subdivision  reaching 
to  the  seconds,  the  same  reduction  will  reduce  the  deci- 
mals of  minutes  into  seconds  jjnd  decimals  of  seconds  j 
thus:   24  h.  62  m.  53,652  s.  =  time  of  arrival  of  John. 

3d  Example.  In  a  besieged  place  the  garrison  consists 
of  2000  men  ;  in  a  retreat  bOO  throw  themselves  into  it, 
to  escape  the  enemy  ;  the  provisions  of  the  place  were, 
sufficient  for  the  former  garrison  for  250  days  ;  how  long 
will  they  last  the  increased  number  of  men,  at  the  same 
rate  of  daily  allowance  ? 

Of  course  the  greater  the  number  of  men,  tlie  less 
time  the  provisions  will  last,  and  that  in  the  inverse 
ratio  of  the  orijjinal  to  the  augmented  garrison  ;  thus 
we  have  the  statement : 

Men.  Men.  Days. 

2000  +  600  :  2000  =  250  :  x 
or  2600  :  2000  =  250  :  a; 

Dividing  by  2000  :  1, 3  :  1   =   250  :  a; 

250 

This  gives  x  = =  192,  3  days; 

1,3 


BXJLE   OP   THSBE.  135 

♦hat  is  :  the  provisions  will  leave  a  small  remainder  after 
192  days,  as  we  obtain  only  three  tenths  of  a  day  over. 

4th  Example.  A  father,  leaving  a  property  of  ^76743, 
makes  the  regulation  in  his  will,  that  it  shall  be  divided 
between  his  two  sons  in  the  inverse  ratio  of  their  ages  ; 
the  one  is  12^  years  old,  the  other  16  and  4  months  ; 
what  will  be  the  share  of  each  ? 

In  this  question  the  inversion  consists  only  in  the  con- 
dition of  the  disposition  itself,  namely  :  that  the  age  of 
the  one  shall  determine  the  share  of  the  other  mutually  ; 
and  the  sum  of  the  ages  forms  the  antecedent  term  of 
the  (comparison  or)  ratio,  given  for  the  proportional 
?hare  of  each  in  the  whole  amount ;  we  have  therefore, 
expressing  the  months  as  twelfth  parts  of  the  year,  the 
following  statement : 

12-|-/_-|-i6-f-^-  :  12-hy6_  c=z  g76743  :  sh.of the  older; 
?2-f  ^2  +  16+^  :  16-1- y\  c=  §76743  :  »'  younger; 
or,  by  successive  reductions  of  each,  which  will  be  easily 
followed  : 

12  -^  I  +  16  -I-  I  :  12  -i-  I  =  §76743  : 


and       12  +  1  -h  16  -f.  f  :  16  -1-  f  = 

76743  : 

or                             28  -h  f  :  12  -M  = 

jj       >j 

and                          28  -h  f  :  16  -H  f  = 

it         a 

or                                           'P'-V  = 

?>              3  5 

and                                         If  3  .  ^B  ^ 

)>              SJ 

lastly.                                       173  :  76  = 

9)          il 

and                                          173  :  98  = 

J)           M 

giving  the  share?   .      ^^  X  76743  _ 
of  the  older      5                 ^^^ 

§33270,  086 

98  X  76743 

■'       younger       = = 

§43432,913 

173 


which  produce  again  the  full  property  within  one  mille, 
lost  by  effect  of  the  interminate  decimal  fractions. 


136  COMPOUND   BUIE    OF  THREE* 

CHAPTER  V. 

Compound  Rule  of  Three, 

§  93.  From  the  principle  explained  in  $  87,  we 
derive,  as  is  tliere  stated,  the  Compound  Rule  of 
Three;  where  several  proportions  being  given, 
which  all  concur  in  the  determination  of  an  un- 
known quantity,  the  product  of  the  different  propor- 
tions term  for  term  being  made,  the  same  principle, 
of  the  equality  of  the  products  of  the  extreme  and 
mean  terms,  takes  jjlace,  as  in  simple  proportion, 
and  the  same  arithmetical  process  gives  the  means 
of  determining  the  unknown  quantity.  It  is  neces- 
sary, of  course,  to  pay  proper  attention  to  the  na- 
ture of  the  ratios  given,  in  respect  to  whether  they 
are  direct  or  inverse,  and  to  make  the  statement  of 
each  accordingly. 

As  the  operation  in  itself  has  already  been  ex- 
plained in  §  87,  and  as  we  shall  immediately  explain 
a  simple  and  general  principle,  by  which  all  such 
compound  influences  and  effects  as  produce  a 
compound  proportion,  or,  what  is  called  the  com- 
pound rule  of  three,  can  be  calculated  with  the 
greatest  ease,  whatever  may  be  tlieir  complication ; 
we  will  here  only  apply  it  to  such  examples  as 
have  for  their  first  ratio  units  of  different  denomina- 
tions, and  form  thereby  what  in  mercantile  calcula- 
tions is  called  the  Chain  Rule,  This  comprehends 
the  finding  of  the  equivalent  of  exchange,  weight 
or  measure,  of  two  places,  by  means  of  the  given 
ratios  of  intermediate  places,  when  the  direct  ratio 
is  not  known.  This  operation  will  exemplify  still 
more  strikingly  the  remark  made  above,  in  relation 
to  the  compensations  of  the  denominations  in  the 
multiplications  and  divisions,  resulting  from  the 
operations  of  the  rule  of  three. 


COMPOUND   RXJIiE    OF   THREE.  137 

Example  1.  li  60lb.  weight  at  Paris,  make  601b.  at 
Amsterdam  ;  451b  at  Amsterdam,  601b  in  New-York  ; 
how  many  pound  of  New- York  make  720  lb  of  Paris  ? 

Multiplying  tliese  proportions,  term  for  term,  we  ob- 
tain the  compound  proportion  by  the  products,  as  below  : 
P.  Am. 


1    :    1  =  60 

:  50 

A.  N.Y. 

1   :   1  =  43 

:  50 

N.Y.  P. 

1      1  =  ^  : 

720 

P.  A.  N.Y. 

Am.  N.Y.  P. 

IXl  X  1    : 

1    X    1  X  1  =60x45Xa;  =  50X60x720 

60X60X720 
1  :  1  =  a;  : 

60x46 
or  X  =  666^666 

by  equality  of  products  of  extremes  and  means. 

The  products  of  the  un'ties  of  the  first  ratios,  give  the 
ratio  of  unity  to  the  product  of  the  second  ratios  ;  the 
denominations  in  the  first  ratios  are  all  compensated,  as 
observed  before,  and  we  obtain,  by  dividing  in  the  se- 
cond compound  ratio  by  the  numbers  multiplying  the 
a',  the  proportion. 

Example  2.  A  merchant  of  Petersburg  has  to  pay  in 
Berlin  1000  ducats,  which  he  wishes  to  pay  in  rubles  by 
the  way  of  Holland  ;  and  he  ha>"  for  the  data  of  his  ope- 
ration, the  following  proportional  values  of  moneys,  viz. 
that  1  ruble  gives  47,5  stivers  ;  20  stivers  make  1  florin  ; 
2,5  florin  make  I  rix  dollar,  Hollandsh  ;  100  rix  dollars, 
Hollandsh  fetch  142  rix  dollars  Prussian  ;  and  finally,  1 
ducat  in  Berlin  is  3  rix  dollars  Prussian  ;  how  man}-^ 
rubles  must  he  pay  ?  This  gives  the  following  statement: 
1  rubl.  :  1  St.     =47,6  :  1 


1   St. 

?  fl.     =     1     :  20 

1  fl.    ■ 

Ird.h.  =     1      :2,6 

1  rd.h; 

Ird  pr=     142:  100 

1  rd.pr  : 

1  due.  =    1      :  3 

1  due.   : 

Irubl.  =   X     :  1000 

12* 

138  COMPOUND   RULE    OF   THREE c 

By  the  sarae  process,  as  in  the  former  example,  is  ob- 
tained : 

1000  X  3  X  100  X  2,5  X  20 

•r  =  . =  2223,87  rub. 

47,5  X    142 

§  94.  In  the  activity  which  nature  presents  to  us, 
as  well  as  in  all  our  actions,  we  observe  this  prin- 
ciple: that  the  product  of  any  cause  into  the  time  of 
its  action  is  equal  to  the  effect  of  it.  Or,  the  product  of 
any  means  whatsoever,  into  the  time  of  tlieir  action, 
or  the  power  which  acts  upon  them,  or  the  conven- 
tional law  of  their  action,  produces  a  determined  ef- 
fect ;  that  is,  it  is  equal  to  it.  Thus  we  have  seen,  that 
a  capital  loaned  on  interest  renders  as  the  product  of 
the  rate  of  interest  into  the  time ;  that  a  man's  la- 
hour  is  the  result  of  the  product  of  his  strength  (or 
power)  into  the  time  he  exercises  this  strength  (or 
power.)  In  all  this  therefore,  we  see  nothing  but 
the  simple  multiplication  of  certain  (actors,  and 
their  product ;  as  has  been  quoted  in  the  remarks 
to  §  71  and  72.  In  the  same  manner  as  products  in 
arithmetic  may  be  tlie  result  of  a  continued  multipli- 
cation, so  may  an  effect  in  nature  be  the  combined 
product  of  a  number  of  causes,  means,  powers,  or 
times;  and  the  effect  itself  may  be  represented  by  a 
combined  product;  as  occurs,  for  instance,  in  high- 
er mechanics,  where  these  quantities  often  appear 
as  multiplied  by  themselves,  or  in  the  square, 
cube,  &c. 

§  95.  If  we  now^  consider  the  relation  of  two  such 
effects ;  that  is  to  say,  their  ratio  to  each  other,  we 
find,  as  we  have  done  in  simple  numbers,  that:  the 
same  ratio  must  take  place  between  the  j)roducts  of 
cause  into  time  (as  it  will  be  siuqilest  to  call  that  by 
a  general  name)  as  that  existing  between  the  effects. 
We  have  now  foj;  some  time  made  use  of  letters  to 
denote  quantities,  before  we  knew  the  numbers 
which  would  correspond  to  them:  we  shall  here 


COMPOTND   UUIE    OF   THREE.  139 

extend  the  advantage  derived  from  it,  in  order  to 
present  this  idea  at  one  glance  in  its  full  connex- 
ions, and  with  the  arithmetical  operations  connected 
with  it.  For  that  purpose  we  shall  designate  the 
objects  of  calculation,  or  the  quantities  of  them,  by 
their  initial  letters,  and  call 

the  cause  =   C^ 

the  time     =   T  I  for  one  of  the  objects  ; 

the  effect   =  E) 

and  for  the  other,  which  is  coin])ared  to  it  in  the 
compound  proportion,  we  sliall  call  the  same  objects 
by  the  corresponding  small  letters,  as : 

the  cause  =  e 
the  time  =  t 
the  effect   =  e 

AVe  then  obtain,  by  the  principles  stated  already 
in  the  remarks  to  §  71  : 

C  X  T  =  E;    and    c  Xt  =  e 
and  for  the  proportion  arising  from  this,  in  a  man- 
ner exactly  similar  to  wliar  lias!  been  done  in  com- 
mon numbers,  we  obtain  the  statement : 

C  X  T:  c  X  t  =   E  :  c 
which  corresponds,  as  simple  products  expressed 
by  their  factors  and  their  results,  to  a  statement 
similar  to 

C         T      c  t  E         € 

3  X  4  :  7  X  9.  =    12  :  63 

It  evidently  follows  from  this,  by  the  division  of 
the  corresponding  terms  of  the  proportion,  that  we 
have  also : 

E      c  E     e 

C  :  c  =  —  :  -    and    T  :  f  =  —  :  - 

T     t  C     c 

And  in  numbers,  also  : 


J 40  COMPOUND   RUIE    OF   THREE. 

12     63  12     63 

3  :7  =  •—  :  —   and   4:9=—:—.    ^ 
4        9  3       7 

§  96,  As  we  have  seen  in  the  preceding  applica- 
tion of  geometric  proportion  to  the  rule  of  three, 
that  whatever  term  of  the  proportion  be  unknown, 
if  the  three  others  are  given,  this  fourth  is  deter- 
mined by  the  principles  of  the  proportion  ;  so  in  the 
present  case,  whatever  may  be  the  quantity  unknown 
in  such  a  compound  rule  of  three,  whether  a  cause, 
a  time,  or  an  effect,  or  a  part  of  the  one  or  the  other 
of  them,  this  quantity  will  be  determined  by  the 
others,  and  obtained  by  tlie  appropriate  mutations 
of  the  proportion,  or  the  operations  of  arithmetic 
resulting  from  it. 

By  this  consideration  and  process  all  the  compli- 
cation, often  resulting  from  combinations  of  direct 
and  inverse  proportions,  in  a  compound  rule  of  three, 
w^hich  are  apt  to  lead  young  calculators  into  mis- 
takes, are  avoided,  because  every  quantity,  in  any 
way  concerned,  is  by  its  nature  placed  as  factor  in 
its  proper  place,  by  the  simple  reflection  of  its  act- 
ing as  eitlier  cause,  time,  or  effect. 

It  may  be  easily  seen  that  it  will  solve  with  ease 
questions  upon  combined  actions  of  capitals  during 
different  times,  as  well  in  interest,  as  in  shares  of 
profit  or  loss,  that  is,  in  partnership,  in  complicated 

*  The  teacher  who  will  take  the  trouble  ti>  speak  with  his  scho- 
lar upon  this  principle,  or  ihe  attentive  reader,  who  willromparo 
it  with  the  circuoistances  that  surround  him,  will  have  no  diffi- 
culty in  explaining  this  simple  idea  ;  its  correctness  and  gene- 
rality will  prove  a  *reat  facility  to  the  intelligent  arithmetician. 
My  own  experience  ha?  proved  to  me  th'.t  it  meets  no  difliculty 
with  boys  of  about  12  or  14  years,  as  scholars  usually  are,  when 
in  common  schools  they  are  thus  far  advanced  in  arithmetic,  and 
that  they  made  the  statements  appropriated  to  it  very  readily,  and 
with  peculiar  satisfa-.tion.  It  furai?hes  the  best  exercise  of  the 
mind  for  the  appropriate  application  of  common  arithmetic.  The 
examples  which  follow  are  worked  out,  and  will,  I  hope,  lead  the 
way  lo  its  proper  and  easy  application. 


I 


COMPOUND    KUIE    OF    THREE.  141 

questions  upon  combined  works,    and  all  similar 
cases,  as  the  following  examples  will  show. 

Example  1.  A  capital  of  §6200  produces  in  6  years, 
at '"7  per  cent.  §2170,  amount  of  interest ;  what  will 
a  capital  of  §9300,  at  4  per  cent,  produce  in  9  j'ears  ? 

Here  the  statement  is  extremely  simple,  thus  : 

C  XT  c  xt=E:e 

G200  X  0,07    X    5  :   9300  X  0,04    x    9  =  2170  :  x 

This  proportion  may  evidently  be  much  reduced.    1st 
by  dividing  by  100,  it  becomes, 
62   X    0,07    X   5  :  93    X   0,04    X    9  =  2170  :  x 
Dividing  by  2, 

31    X   0,07    X    5  :  93    X  0, 02  x  9  =  2170  :  x 
Dividing  by  70, 

i    31    X    0,001    X  5  :  93  X  0,02  X  9  =    31   :  re 
Dividing  by  31, 

0,001     X     5  :     93   X    0,02    X    9  =      1   :  x 

93   X    0,02    X   9         16,74 

Giving,  X  = = =  §3348 

6   X  0,001  0,005 

That  is,  the  capital  of  §9300,  at  4  per  cent,  produces,  in 
9  years,  ;gf3348  interest. 

Example  2.  A  capital  of  ^9500,  at  6  per  cent,  inter- 
est, annually  produced  ^4560,  m  8  years,  at  what  rate 
of  interest  must  a  capital  of  ;^  12000  be  lent  out,  which 
shall  render  §4800  in  5  years  ? 

Statement  : 
9500   X   0,06    X   8   :   12000  x   5  X  a;  =  4560  :  4800 
Reducing  as   above,  by  dividing  the  first  by  500,  and  the 
second  by  40  ; 

19   X   0,06    X   8  :  24   x    5   X    a;  =   114  :   120 
Dividing  the  first  and  third  term  by  6  ; 

19   X    0,01    X   8  :  24    X    5    X   a;  =   19  :   120 

Dividing  the  first  and  third  by  19,  and  the  second  and 
fourth  by  24. 

0,01    X    8  :  a;   X  5  =  1   :  6 


142  COMPOtTND  RULE   OF  THREE. 

Dividing  the  second  and  fourth  term  by  6,  and  executing 
the  multiplication  indicated  in  the  first  term,  we  obtain  : 

0,08   :  rr  =   1    :   1 

Or,  the  rate  per  cent.  =  a;  =  0,  08  or  8  per  cent. 

Thus  the  simple  reductions  of  the  proportion  given, 
has  furnished  the  result.  It  is  evident,  that  if  we  had 
at  the  first  outset  of  this  and  the  preceding  example,  ex« 
pressed  the  term  in  which  x  is,  by  the  other  three,  we 
would  have  reached  the  same  results  by  the  compensa- 
tions in  the  numerator  and  denominator,  and  the  factors 
of  X  with  the  opposite  numerator. 

Example  3.  Two  men,  in  partnership,  contribute  as 
follows  :  A  puts  in  7521  dollars,  which  he  withdraws  af- 
ter 5  years  and  a  half  B  puts  in  9772  dollars,  which  act 
in  the  company  during6  years,  before  which  time  the  ac- 
counts cannot  he  settled.  It  is  required  to  determine  the 
share  of  each  in  the  general  result  of  all  the  operations, 
(which  are  taken  together,)  amounting  to  a  net  profit  of 
15472  dollars  ? 

The  sum  of  the  products  of  the  stocks  into  the  times 
of  their  acting,  are  here  to  be  compared  to  each  single 
product  of  stock  into  the  time  of  its  acting,  as  cause 
and  time  ;  the  whole  benefit  evidently  represents  the 
efiect,  corresponding  to  the  whole  stock,  and  its  time  of 
action. 

Thus  we  obtain  the  two  following  statemens  : 

7521  X5,6+9772X6  :  7521  X5,5  =  15472  :  share  of  A 
7521X5,5+9772x6  :  9772x6  =  15472  :  share  of  B 
Or  99997,5  :  41365,5  =  15472  :  share  of  A 
And  99997,5  :  58632,0  =  15472  :  share  of  B 

Here  we  evidently  obtain,  as  in  the  case  of  a  bank- 
rupt, treated  in  a  former  example,  a  constant  fraction 
from  the  third  term  divided  by  the  first,  with  which  the 
second,  or  the  product  of  the  stock  into  the  time  of  each 
partner  is  to  be  multiplied,  to  obtain  his  share  in  the  pro- 
fit ;  or  we  have  ; 

15472 

Theshareof  A  = X  41365,5  == 

99997,5 


COMPOtrWD    RULE   OF   THREE. 

15472 

The  share  of  B  = X    58632  = 

99997,6 

Example  4.  If  9  men  working  6  days,  at  the  rate  of 
8  hours  per  day,  can  build  a  wall  of  152  feet  long,  and  9,5 
feet  high,  how  many  days  must  16  men  work,  at  the 
pate  of  10  hours  each  day  to  build  a  wall,  295  feet  long, 
and  17,5  feet  high? 

Example  5.  If  180  men,  working  6  days,  each  day  10 
hours,  can  dig  a  trench  of  200  yards  long,  by  3  yards 
wide,  and  2  yards  deep,  how  many  days  will  100  men 
take  to  dig  a  trench  of  360  yards  long,  4  wide,  and  3 
deep,  by  working  8  hours  in  a  day  ? 

This  gives  the  following  statement,  in  which  the  effect 
is  a  compound  product,  because  the  trench  has  the  three 
dimensions  of  length,  breadth,  and  depth.  The  reduc- 
tions which  it  admits,  will  here  be  made  without  men- 
tioning them,  under  the  supposition  that  the  preceding 
examples  have  shown  the  principle  of  them  ;  y  being 
taken  for  the  unknown  days. 

180  X  lOX  6  :  100  X  8X  y  =200x3X2  :  360x  4  x  *" 
18   6:  X8  X  y  =  10  :  36 
9  X  3  :  2  X  y  =   1  :  3,6 
27  :  2/  =  1  :  1,8 

2/  =  27  X  1,8  =  48,6  days. 
Example  6.  A  hare  is  50  leap?  before  a  greyhound, 
and  he  takes  4  leaps  while  the  greyhound  takes  3  ;  but 
2  greyhound's  leaps  are  equal  to  3  hare's  leaps ;  how 
many  leaps  must  the  greyhound  make  to  overtake  the 
hare  ? 

This,  as  it  appears  a  standing  question  in  all  books  on 
arithmetic,  is  well  adapted  for  an  example  in  this  case. 
The  proportion  of  the  leaps  as  given,  are  : 

In  time  ;      hare's  leap  :  hound's  leap  =  4:3 
In  length ;  "  :  "  =2:3 

The  compound  ratio  of  them,  or  the  product  of  cause 
into  time,  which  determines  the  efiect,  is  therefore  : 

hare  :  hound  =  8:9 


144  GEJfERAL   APPtlCATIOX   OF 

If  we  call  the  distance  the  oound  has  to  run  =  x,  in 
hare's  leaps,  (as  the  determined  distance  is  given  in  this 
kind  of  quantity,)  the  hare's  ron  in  the  same  time  will  be 
X  —  50  in  the  time  they  both  run  ;  these  two  circum 
stances  of  the  data  give  the  following  statement, 
a;  :  a;  —  60  =  9  :  8 

By  comparing  the  antecedent  with  the  difference  be- 
tween antecedent  and  consequent,  we  obtain  : 

a-  :  60  =  9  :  1 
X  =  9  X  60  =  450  hare's 

As  the  hare's  leaps  are  |  of  the  hound's,  this  distance 
will  require  300  hound's  leaps  ;  so  many  therefore,  he 
will  have  to  make  to  overtake  the  hare. 


CHAPTER  VI. 

General  Application  of  Geometric  Proportion* 

§  97.  When  two  or  more  proportions  are  given, 
two  unknown  quantities  may  be  determined  by 
means  of  the  mutations  of  these  propoi-tions ;  and 
the  determination  of  the  one  by  the  other,  appropri- 
ating the  choice  of  the  operations  to  the  given  case^ 
in  such  a  manner  that,  by  whatever  operation  the 
quantity  sought  is  involvetl  with  other  given  quanti- 
ties, these  become  disengaged  by  performing  the  con- 
trary operation ;  this  is  grounded  upon  tlie  principle 
of  arithmetic  stated  in  the  beginning,  that  each  ope- 
ration (or  rule  of  arithmetic)  has  its  opposite  opera- 
tion ;  and  this  is  the  principle  used  in  all  the  reduc- 
tions that  have  been  made  in  the  proportions  in  the 
preceding  sections,  to  obtain,  or  render  easy  the  ob- 
taining of,  the  results. 

1st  Example,  Two  numbers  are  in  the  ratio  of  2  :  3  ; 
when  each  is  augmented  by  4,  they  are  in  the  ratio  of 
5:7:  what  are  these  numbers  ? 


GEOMETRIC   PROPORTION".  145 

Denoting  the  one  by  x,  the  other  by  y,  we  have  the 
first  statement : 

2:3  =  X  :  y 

And  as  the  fourth  term  is  equal  to  the  product  of  the 
two  mean  terms  divided  by  the  first,  we  have  slso : 

3Xx 
2:3  =  x : 

2 

3Xx 
that  is,  y  =  

2 

The  second  proportion,  by  using  this  result,  will  be 
stated  thus : 

3XX 
5  :  7  =  X  +  4  : +  4 


Multiplying  the  second  ratio  by  2 : 

5:7   ==  2x  +  8:3Xx  +  8 
By  subtracting  antecedents  from  consequents  : 

5  :  2  =  2x  +  8  :  X 
Subtracting  consequents  from  antecedents  twice  c 
3  :  2  =  X  4-  8  :  X 
1  :  2  =  8  :x 
whereby  a;  =  2  X  8  =  16 

And  y  by  the  first  proportion,  placing  the  value  of  x, 
J  ast  found,  in  its  place : 

2  :  3  =   16  :  y 
or  1  :  3  =     8  :  y 

whence  2/  =  3  X  8  =  24 

2d  Example.  A  father  being  asked  how  many  sons  and 
daughters  he  had,  answered,  "  If  I  had  two  more  of  each, 
I  should  have  three  sons  to  two  daughters,  and  if  I  had 
two  less  of  each,  I  should  have  two  sons  to  one  daugh- 
ter ;"  how  many  sons,  and  how  many  daughters,  had  he  ? 

This  evidently  furnishes  two  proportions,  one  stated  by 
the  sums  of  the  numbers  sought  and  2,  and  the  other 

13 


146  GENERAL   APPXTCATIOIf   OF 

by  the  difference  between   the  numbers  sought   and    t 
in  the  other,  as  follows  : 

Calling  the  number  of  the  sons  =  x  ; 

That  of  the  daughters  =  y  : 

a-  4-  2  :  2/ H-  2  =;  3  :  2 
x~2:?/~2   =  2:l 
From  these  proportions  are  obtained,  by  steps  ground- 
ed upon  the  principles  of  proportion,   demonstrated  in 
§  86,  the  following  buccessive  results  : 

From  X  -\-  '2  :  y  -\-  2  =  3:2 

x-^2:x-\-2  —  y— 2  =  3:    1 
X  -\-  2  :  X  -y  =z  3  :  I 
In  like  manner,  from 

ar  —  2:y  —  2  =  2:1 
a;  -  2  :  ^  -  2  -  j^  +  2   =^  2  :  1 
X  —  2:x-.y  =  2:l 
Dividing  these  two  results  term  by  term,  as  hy  §  86  : 
x-{-2  3 

:  1    =   -.  :  I 

a:  —  2  2 

or  a:-f2:a:_-2=--3:2 

From  this     x  +  2  +  x  ^  2  :  x  +  2  -^  x  J{^  2  =  b  :  \ 
or  2  ^  :  4  ==  5  :  1 

and  X  ',2  =.  b\\ 

a:  =  2  X  5  =  10  the  number  of  sons. 
Though  this  determines  the  number  of  daughters,  if 
we  place  this  value  in  eitfaer  one  of  the  first  proportions, 
and  then  determine  the  y^  as  in  the  foregoing  example  ; 
still  it  is  evident  that  both  x,  and  ?/,  are  dependent  upon 
the  data  in  exactly  the  same  manner ;  1  will  therefore 
also  determine  y  by  a  similar  ai>propriate  process,  as 
it  will  be  a  good  ex.i  -pie  to  show  the  princijiles  of  this 
use  of  proportions  in  determining  quantities  in  general. 
We  made  the  first  term,  containing  ^,  our  standing 
term  ;  we  shall  have  now  to  make  the  second  terro> 
containing  ?/,  the  standing  term  of  the  operation.  Thus 
we  have  from  the  first  proportion: 


GEOMETRIC  PROPORTIOy. 


147 


r-h2-?/-2:y  +  2  =   1  :2 
or  X  —  y  :  y  -\-  2  =   1  :  2 

And  from  the  second  proportion  : 

a:  —  2  —  y4-2:y— 2  =   1;1 
X  —  y  :  y  —  2  =    1  :  1 
Dividing  these  two  proportions  term  by  term,  as  be- 


lore,  we  obtain  : 


y  +  2 


=   1 


y-2  I 

or  y  — 2:2^4-2=1:2 

tBy  sum  and  difference  ; 
2/  +  2  +  y-2:2/-t-2~»/  +  2  =  3:1 
or  2  y  :  4  =  3  :  1 

y  :  2  =  3  :  1 
giving    y  =  2  X  3  =  6    for  the  number  of  daughters. 

3d  Example.  I  asked  my  two  neighbours,  John  and 
Peter,  how  many  head  of  cattle  each  had  ;  Peter,  thinking 
to  puzzle  me,  says,  '*  Our  cattle,  taken  together,  are  to 
what  John  has  more  than  I,  in  the  ratio  of  3  to  2  ;  and 
if  we  multiply  the  two  numbers  of  our  cattle  together, 
that  product  will  bd  to  all  our  cattle  in  the  ratio  of  5  to 
3.  I  find  how  much  each  of  them  has  in  the  following 
way. 

Calling  John's  cattle  ===  x 
and  Peter's  cattle  =  y 
the  first  proportion  given  furnishes  me  the  statement 

X  -^  y  :  X  —  y   =  3:2 
and  the  second, 

X  -^  y  :  X  .  y  =  3:6 

By  addition  and  subtraction  of  the  first  proportion  Is 
obtained  ; 

X'{-y'\^x-'y:x+y-x  +  y  =  5:  1 


or 

2a:  :  2y  =  6 

:  1 

X  :  y  s=  5 

1 

thence 

«-f  2/:y  =  6 

1 

and 

x  +  y  :  a:  =  6  • 

d 

148  GENERAL  APPLICATION  OP 

Diriding  the  second  proportion  given  by  either  of 
these,  term  for  term,  I  get  : 

x-i-y     X  .y  3  1 

x  +  y        y  6  2 

a:-^1/     X  .  y  3  1 

X  -^  y       X  6  2 

tha^-is,  1  :  a:  =  3  :  30  =   1  :  10 

and  1  :  y  =  i  :  2 

giving  a;  =  10  ;  y  ~  2 

So  I  find  John  has  10  head  of  cattle,  and  Peter  appears 

lo  be  richer  in  puzzles  than  in  cattle,  which  he  did  not 
like  to  tell  me. 

4ih  Example.  A,  B,  and  C,  in  a  joint  speculation, 
gain,  and  give  only  the  following  account  of  the  quantity 
each  gained  :  the  product  of  the  gain  of  A  into  that  of 
B  is  equal  to  gl200,  that  of  A  into  that  of  C  =  gl800, 
and  that  of  B  into  C  =  §2400 ;  what  was  the  gain  of 
•ach  ? 

This  example  will  show,  that  an  equality  of  products 
as  is  given  here  expresses  a  geometric  proportion  equally 
as  well  as  an  equality  of  fractions  or  ratios;  for  by  the  de- 
composition of  these  products  into  the  extreme  and  mean 
terms  of  a  proportion,  we  obtain  the  three  proportions  : 
a  :  40  ==  30  :  y 
X  :  60  =  30  :  z 
y  :  40  =  60  :  z 
Dividing  the  first  by  the  second,  term  by  term,  we 
obtain : 

X     40         30     y 

X     60         30     z 
or       60  :  40  =  r  :  y 
Dividing  this  proportiott  by  the  third,  term  for  term.: 
60     4a"       ■  z      y 

V      40         60     r 


GEOMETftlC    PROPOHTIOIf. 

60  :y  =  z  Xz  :  60y 
60  :  z  =  z  :  60 
z  =  60 

DiFiding  the  first  and  third,  term  by  term  : 

X     40         30     y 

y     ,40  60     z 

X  :  y   =s   30  2r  :  60  ^ 
X  :  1   =  z  :  2 

Multiplying  this  by  the  second,  term  for  term  : 
xXa;:60  =   30X2':2e 
a:  :  30  =  30  :  a; 
a;  =  30 

Dividing  the  second  by  the  third,  term  for  term  : 
a:     60         30      z 


149 


y     40         60      z 

40  ar  :  60  2/  =='  30  :  60 
4a;  :  7/  =  3  :  1 
a;  :i^  =  3  :4 

riding  this  by  the  first,  term  for  term  : 
X     y  14 

X     40         10     y 
or  40  :  y  =  y  :  40 

2/  =  40 

This  example,  expressly  chosen  for  its  simplicity,  may 
suffice  to  explain  the  principle. 


13^ 


PART   IV. 

EXTENSION  or  ARITHMETIC  TO  HIGHER  BRANCHES 
AND  OTHER  PRACTICAL  APPLICATIONS. 


CHAPTER  I. 

Of  Square  and  Cube  Roots. 

§  98.  When  in  a  multiplication  the  two  factor* 
are  equal,  the  product  is  called  a  square  ;  because 
it  corresponds  to  what  would  be  produced  in  nature 
by  laying  off  the  quantity  which  these  numbers  repre- 
sent, in  any  unit  of  lineal  measure,  in  two  directions 
perpendicular  to  each  other;  and  completing  the 
figure  by  two  equal  lines,  drawn  perpendicular  at 
^  the  end  of  these ;  as,  for  instance,  taking 
4  feet  and  laying  them  off  upon  AB,  and 
also  upon  AC,  and  then  drawing  BD,  and 
CD,  at  equal  distances,  again  perpendi- 
cular to  AB,  and  CD;  ABCD  will  be  a 
square,  representing  the  square  of  A, 
E  F   that  is,  4  X  4  =  16. 

(ICIIi         The  product  of  any  two  numbers  may 
A     be  represented  in  the  same  way,  by  two 
ij     lines  perpendicular  to  each  other,  divided 
into  equal  parts,  and  completing  the  rec- 
tangular figure,  having  its  opposite  sides 
equal ;  as  here  the  figure  EFGH, 
"^  "        So  we  may,  when  we  have  sucli  a  sur- 

face, or  product,  given,  and  one  of  the  sides,  find 
the  other  side  by  division,  as  is  evident  from  the 
second  figure.     But  when  the  figure  is  a  square,  as 


\± 


BitrARB    AND    CUBE    ROOTf. 


151 


in  the  first  case,  we  can  find  the  two  equal  sides  of 
it  by  a  peculiar  process,  which  is  called  the  extract 
(ion  of  the  square  root;  the  principle  of  which  it  is 
now  intended  lo  explain. 

For  this  purpose  it  is  necessary  to  investigate 
what  a  product  is  composed  of,  by  decomposing 
each  factor  into  two  parts,  not  unlike  the  method 
we  have  used  to  show  ihe  propriety  of  the  principle 
of  carrying  in  multiplication;  namely,  we  divide 
the  number  into  two  parts;  thus,  for  instance,  we 
would  write  14  as  10  -f-  4 ;  or  merely  consider  it 
as  so  composed,  and  by  multiplying  tlie  number 
into  itself  under  that  form,  keeping  each  individual 
result  separate,  we  shall  obtain  the  following  pro- 
cess and  results : 

14 
14 


4X4 


4  X  10 
10  X  10 -f  4  X  10 


10  X  10 -f  2  X  4  X  10  +  4  X  4 
100  +  80  +  16  =  196 

ITiat  is,  we  obtain  by  the  product  of  the  units 
4X4  =  16  ;  by  the  product  of  the  unit  of  the  mul- 
tiplier into  the  tens  of  the  multiplicand,  4  x  10  =  40, 
and  the  same  again  by  the  product  of  the  tens  of 
the  multiplier  into  tlie  units  of  the  multiplicand ; 
then  lastly,  by  the  product  of  the  tens,  10  X  10 
~  100. 

Tliis  gives,  by  the  addition,  three  distinct  pro- 
ducts, viz  : 

1st.  The  square  of  the  first  part,  that  is,  the 
product  of  the  first  part  into  itself,  here  10  X  10. 

2d.  Twice  the  product  of  the  two  parts  into  each 
other,  here  twice  4X10,  or2X4XlO. 


152  SI^UAKE    AND    CUBE    ROOTS. 

3(1.  The  square  of  the  last  part,  or  the  liiiits, 
licre  =i:  4  X  4.  '■''''  '*^-'"'  "  s  "■  •. 

In  making  the  cUvision  of  the  number  acfcrtrding 
to  our  decimal  system  of  numeration,  they  follow 
the  same  order  in  magnitude  as  here  stated.  We 
find  also  by  the  ins])ection  of  tliis  result,  as  we  know 
besides  by  the  multiplication  table,  that  the  product 
of  the  units  can  iiitluence  two  places  of  figures, 
namely,  units  and  tens,  and  cannot  influence  the 
third ;  the  same  is  the  case  with  any  of  the  subse- 
qjient  numbers,  each  influencing  only  tfie  rank 
tvhich  it  occupies,  and  t!)e  next  higher  rank ;  this 
gives  the  principle,  by  wliich  we  may  know  in  any 
number,  of  how  many  numbers  the  square  root  will 
be  composed,  namely  :  by  dividing  it  into  as  many 
pairs  of  figures,  from  tlie  rig]»t  hand  side  towards 
the  left,  as  it  will  admit ;  the  immbor  of  these  divi- 
sions, will  be  the  number  of  figures  of  the  square 
root. 

As  tlie  extraction  of  the  square  root  of  a  number 
will  again  be  the  opposite  of  tlie  elevation  to  the 
square,  the  above  operation  must  be  executed  in  an 
inverted  order  to  extract  the  square  root,  as  in  di- 
vision the  inverse  order  of  the  multiplication  has 
been  followed. 

The  operation  of  raising  to  a  power  is  also  called. 
Involution,  and  the  extracting  of  the  root.  Evolution, 

In  order  to  denote  in  an  abridged  manner  the 
jnultiple  of  a  number  by  itself,  the  idea  will  readily 
occur,  to  write  the  number  only  once,  and  to  indi- 
cate the  number  of  factors  intended,  by  jflacing  a 
small  number  at  the  top  and  to  the  right  hand  of  the 
number,  corresponding  witli  this  number  of  factors ; 
and  103  =  10X10;  i  03  =  ioxiO):<10;  and  so  for 
any  other.  To  indicate  the  extraction  of  the  root  the 
sigh  ^fOv  an  extended  r,  is  written  before  the  num- 
ber; as  v/  196,  denotes  the  square  root  of  196;  if 


SCtUARE  AND  CUBE  ROOTS.  15$ 

other  roots  are  to  be  extracted,  the  number  corres- 
ponding to  the  degree  of  the  root  is  written  in  the 
-/,  as  ^;  ^;  and  so  on;  hut  a  much  better  method 
is,  to  continue  the  same  manner  of  notation  as  in 
raising  numbers  to  their  powers,  expressing  the 
roots  in  their  corresponding  fractions,  so  that 
^  196  =  (196)*,-  ^196  =  (I96)i;  and  so  on  in 
higher  degrees. 

§  99.  In  Evolution  the  first  step  will  therefore 
be,  as  in  divisinn,  to  find  that  number  which,  mul- 
tiplied into  itself,  will  give  the  product  nearest  be- 
low the  most  left  hand  number;  this  square  being 
subtracted,  the  remainder  must  furnish  the  two 
other  products ;  as  the  second  of  these  is  the  larger, 
if  wc  multiply  tiie  number  found  before  by  2,  and 
divide  the  remainder  by  it,  we  shall  have  a  number 
as  quotient,  near  the  second,  or  next  following  num- 
ber ;  with  which  we  shall  then  have  to  execute  the 
two  products,  indicated  by  the  above  result  of  such 
a  multiplication. 

lit  Example.  Let  the  above  number  be  chosen  to  ei- 
tract  the  square  root ;  to  explain  tl^§  direct  inversion  of 
the  operation,  or  to  execute  '  V' '/  ~ 

^  1196  =  10 -f- 4  =  14 
First  square  10  X  10  =       1  00 

Remainder  =  96 

;Divisor  2  X  10  =a  20)   in  96  ;  4  times 

(20  -I-  4)  x4  =  96 

No  remainder  00 

The  number  divided  off  by  2  from  the  right  hand 
shows  that  the  root  has  two  places  of  figures  ;  so  the  first 
will  be  in  the  tens,  and  the  number  in  the  second  division 
being  l,the  square  root  of  which  is  also  1,  the  first 
•quare  will  be  10  X  10  =  100 ;  the  root  10  being  writ- 


154  SCtr\TlE  AXD  CUBE  ROOTS. 

ten,  the  square  =  100,  is  subtracted  from  the  whole  196  ; 
the  remainder,  96,  being  written,  the  divisor,  which 
shall  serve  to  tind  the  other  number,  will  be  the  product 
2  X  10  =  20  ;  which  being  found  to  go  4  times  in  the 
remHinder,  Tis  written  in  the  root,  and  being  also  added 
to  the  20,  tlii;  sum  ot  both  is  multiiilied  by  4  again,  as  we 
have  found  it  to  be  factor  in  both  the  two  last  terms  ;  the 
product  of  this  =  96,  written  under  the  remainder  96, 
being  exactly  equal,  gives  the  14  as  the  square  root  of 
196  in  return. 

2d  Example,  Let  it  be  given  to  extract  the  square  root 
of  a  number  of  more  than  two  places  of  figures,  as  13456. 
Dividing  the  jiumber  off  as  before  directed,  we  find,  that 
the  root  must  have  three  jdaces  of  figures,  or,  the  first 
figure  will  be  in  the  hundreds  ;  thus  we  obtain  the  fol- 


low 


mg  process ; 


v/1 
100X100       J 


'H) 


Divisor  2  X 100' 
Product  2X100-1- 10 
multiplied  by  10 

Remainder 

Divisor  =  2X110 

(220-1- 6) X6  =5 


56  =  100-^10+6  =  116 


34 

oolilyo 


66     c^uotient  =  10 
00 


56 


56 


quotient  =  6 


0  00 


It  will  easily  be  conceived,  that  here,  as  in  division, 
every  number  does  not  give  a  whole  number  for  a  radi- 
cal, because  every  number  is  not  the  product  of  another 
number  multiplied  into  itself. 

"We  may  evidently  in  large  numbers,  by  way  of 
abridgment,  take  only  the  two  next  numbers 
down,  as  in  division,  arid  consider  the  former  as  a 
ten,  in  relation  to  this  number  taken  down,  and 
proceed  thus  to  the  end,  or  to  any  desired  number 
of  places  of  decimals;  for  the  process,  as  first 
mentioned,  will  proceed  in  this  case  according  to  th# 


^^^^H  S^UABE  AND  CUBIS  HOOTS.  155 

^^^^stem,  exactly  as  if  it  was  a  mere  division 
continued  to  decimals,  only  the  mode  of  making  up 
the  successiye  products  Avhich  are  to  be  subtracted 
being  different.  Therefore,  also,  the  evolution  of 
a  number  with  decimal  fractions  is  exactly  the  same 
as  the  evolution  of  whole  numbers,  whether  it  have 
an  exact  root  or  not.  But  it  must  be  remarked, 
what  the  principles  upon  which  decimal  fractions 
arc  grounded  might  easily  suggest,  namely :  that 
the  partitioning  into  pairs  must  again  begin  from 
the  unit,  botij  to  the  left  and  to  the  right;  there- 
fore, if  there  be  an  odd  number  of  decimal  places, 
a  0  must  be  placed  to  the  right  to  make  up  the  pair, 
Avhich,  as  is  well  known,  does  not  cJiajige  the  value 
of  the  fraction. 

The  following  two  examples  will  suffice  to  give  a  cor- 
rect idea  of  it,  and  lead  to  the  practice  of  this  operation. 

To  execute  ^  1419,7064,  being  a  number  with  a 
decimal  fraction:  writing  it  partitioned  off  as  before,  the 
foUowins  results  : 


First  square 

Divisor  2  X  30 

(60  -i-  7)  7 

Dirisor  2  X  370 

(740  ■\-^)^ 

Divisor  2  x  37,  6 
(7520  -f  ^)  8 


9 

19, 

5 

19 

4 

69 

50 

44 

6 

B 

6 

0 

7C 


24    =37,68 


quot  = 


0224 


00  00 


quot  =  G 


24     quot  =  8 


^  Here  the  process  is  evident,  from  the  ei^reSsions 
placed  opposite  to  each  number ;  the  number  obtained 
is  always  augmented  by  a  0,and  multiplied  by  2,  to  form 
the  divisor,  which  from  the  remainder  gives  the  next 
figure  ;  this  is  considering  it  as  the  ten  of  the  following 
number  ;  the  quotient  added,  and  the  sum  multiphed  by 


156  5(tUAIl£  XTUD  CUBE  ROOTS. 

it,  gives  the  product  to  be  subtracted  ;  and  the  remainder 
is  to  be  treated  as  before. 

Exactly  in  the  same  manner  the  following  example 
gives  v^  2  ;  it  is  here  placed  without  any  further  indica- 
tion, in  order  to  give  room  for  study. 

^21       =   1,41421356  4*  &c. 

1 


1  oc 

9i 

) 

! 

4 

2 

00 

81 

.1 

1 
1 

19 
12 

00 

96 

1 

-1— 
6 
5 

04 

65 

64 

t 

38 
28 

36 

28 

00 
41 

10 

8 

1 

07 
48 

59  00 
52  69 

7 

1 
1 

59 
41 

06  31 
42  13 

00 

25 

-I 

17 

64117 

75 

00 

^5  99.  From  the  preceding  we  have  only  a  short  and 
easy  step  to  make,  by  means  of  reflections  grounded 
upon  the  principles  just  used  to  explain  the  extraC" 
tion  of  square  roots,  in  order  to  determine  the  prin- 
ciples upon  which  a  quadratic  equation  is  solved; 
that  is,  to  furnish  the  means  to  determine  an  un- 
known quantity,  which,  in  a  combination  with  oth- 
ers, would  be  multiplied  into  itself,  or  what,  as  wc 
have  stated  above,  is  said  to  be  squared.     To  make 


SCtVABE  AfTD  CUBE  HOOTS. 


15T 


tbetjxplanation  more  simple,  we  may  use  two  means, 
which  taken  in  conjunction  will,  I  hope,  leave  to  the 
attentive  student  of  this  hook  no  difficulty. 

I  wish  to  introduce  this  here,  although  unusual, 
because  its  absence  would  leave  us  in  the  subsequeni: 
parts,  when  we  shall  treat  of  progressions,  without 
the  means  of  finding,  or  satisfactorily  explaining, 
the  solution  of  certain  questions  arising  from  them ; 
for  I  have  proposed  to  myself,  never  to  lead  the 
student  blindfold  over  any  step ;  while  at  the  same 
time  I  wish  to  give  him  all  the  means  of  calcu- 
lation in  arithmetic,  that  he  may  desire,  in  a  man- 
ner satisfactory  to  a  reflecting  mind. 

We  have  before  decomposed  the  number,  of  which 
w^e  wished  to  show  the  different  products  forming 
the  square,  into  two  parts,  and  have  there  shown, 
that  tlie  square  number  resulting  was  composed  of 
the  sum  of  the  squares  of  the  two  parts,  and  twice 
the  product  of  the  two  factors  into  each  others 
we  there  decomposed  the  14  into  10  and  4 ;  we 
chose  this  division  on  account  of  its  direct  appli- 
cation to  the  extraction  of  the  square  root  of  a 
number  written  in  our  usual  decimal  system,  but 
Any  division  will  do  the  same  thing. 

If  in   the    annexed   fi-  ^  _^ 

gure,  of  14  subdivisions 
on  each  side,  we  di- 
vide the  sides  into  9  and  5 
parts,  the  result  will  be 
exactly  the  same ;  we  shall 
have  tlie  square  d9a9  =■ 
9  X  9  =  81 ;  the  product  of 
5X9  twice,  on  each  side  of 
this  square,  in  9abBf  and 
9acC,  and  the  small  square 
ubDc  =  5X5,  which  toge-  J 
ther  will  fill  up  the  large 
square  JBDC^  and  summing  up  these  products.* 
14 


;  M    ■  M    M^ 

I  I    I    I  I    I    I    I 

I  I    I    I  I    I    M 

I  I    M  I    I    I    I 

I  I    I    I  I    M    f 


B 


158  SC^lTABE  ANi)  ttlBM  ROOTS. 

obtained  by  the  multiplication  as  above,  we  of  course 

f    81  "^    exactly    as    by   the   othei     divi- 

[      90    I    sion.     In  like  manner  any  other 

obtain  -^      55    ^  division    would    give    the    same 

I I    sum. 

L  196  J  As,  therefore,  a  square  number 
can  be  decomposed  in  any  two  parts,  so  as  to  obtain 
from  it  two  smaller  squares,  and  twice  the  product 
of  the  two  parts  into  each  other,  we  are  allowed  to 
consider  any  square  number  to  be  thus  composed. 

We  have  seen  in  the  very  beginning,  that  in 
aritlimetic  we  have  always  two  operations,  exactly 
opposite  to  each  other,  the  one  always  compen- 
sating the  effect  of  the  other.  We  have  seen 
in  treating  of  proportions,  that,  when  the  same 
operation  was  executed  on  both  sides  of  the  sign  of 
equality,  the  results  were  again  equal,  and  therefore 
the  principle  of  equality  still  subsisted ;  or,  what 
is  the  same,  that  equal  operations  performed  upon 
equal  quantities  do  not  destroy  the  equality;  by 
this  means  we  were  enabled  to  (obtain  solutions  of 
questions,  or,  what  is  the  same)  determine  unknown 
quantities,  variously  involved  by  other  known  ones. 
If  now,  in  application  of  these  principles,  we  con- 
sider an  unknown  quantity  in  any  manner  involved, 
which  appears  in  any  one  or  more  of  the  parts, 
multiplied  into  itself,  that  is,  in  the  square,  and  in 
other  parts  simple,  we  are,  by  the  principle  last 
shown,  authorised  and  enabled  to  separate  the 
square  from  all  other  numbers,  or  quantities ;  and 
we  can  consider  it  thus  insulated,  according  to  the 
explained  principles  of  the  division  of  the  square, 
as  representing  the  square  of  the  first  part  or  sub- 
division of  the  number,  or  of  the  square. 

To  apply  this  to  an  example,  we  must  again  give 
our  unknown  quantity  a  designation,  and  treat  it  as 
if  we  knew  it,  until  it  comes  to  stand  alone  on  one 
side  of  the  sign  of  equality,  which  gives  tlie  solu- 


I 


SttTTARE  AND  CTJBE  BOOTS*  159 

tion,  by  indicating  that  it  is  equal  to  the  result  ol 
the  combination  represented  by  the  known  quanti- 
ties on  the  other  side  of  the  sign  of  equality.  Then 
the  terms  multiplied  by  the  unknown  quantity  must 
be  considered  as  representing  twice  the  product  of 
the  first  term  into  the  second,  or,  in  that  case,  of 
the  unknown  quantity  into  the  known  ones.  The 
half  of  tliis  factor  being  squared  will  represent 
the  smaller  square ;  (or  in  general  the  other  square 
needed  to  complete  the  entire  square;)  by  the  addi- 
tion of  this  square  on  both  sides  of  the  equality,  a 
square  number  is  obtained,  of  which  the  square  root 
can  be  exti'arted  by  the  rules  given,  or,  what  is  in 
this  case  equivalent,  which  can  be  expressed  by  the 
given  numbers.  The  quantity  sought  for  is  there- 
fore known  from  it. 

Example.  Suppose  we  had  given,  by  the  result  of  a 
calculation,  a  combination  of  quantities  which  have  the 
following  form; 

480  =  3x2  -f  36x 

160  =  x3  +  12x 

196  =  xa  +  12r-l-36 
v/  196  =  X  -1-  6 
14_  6  =  8  =x 
having  the  x  in  the  square  multiplied  by  3  ;  this  must 
first  be  disengaged,  by  dividing  both  sides  by  3  ;  this  gives 
the  second  line  ;  then  the  12,  multiplying  the  simple  x, 
represents  the  product  of  2  into  the  second  part  of  the 
subdivision  of  the  whole  square  ;  therefore  its  half,  or 
6,  is  the  side  of  this  second  square,  when  x  is  the  side 
of  the  other,  because  the  12x,  or  2X6Xx,  must  re- 
present the  double  product  of  the  two  parts,  like 
9  abB  -f  9acC.  If,  therefore,  we  square  the  6,  and  add 
it  to  both  sides,  by  which  the  equality  is  not  changed,  we 
shall  have  on  the  right  hand  side  a  full  square,  in  which 
the  X  is  the  side  of  one  of  the  lesser  squares,  and  the 
other  is  known  ;  thus  the  third  line  above  is  obtained  ; 
the  two  parts,  into  which  the  square  appears   divided. 


160  S(IUARE  AND  CUBE  HOOTS, 

are  therefore  x  and  6,  which  will  together  be  equal  t& 
the  square  root  of  196  ;  this  gives  the  fourth  line  Ex- 
tracting the  squire  root  of  196,  gives  14,  and  if  the 
6  is  subtracted  on  both  sides,  gives  the  value  of  x  as  in 
the  last  line,  for  the  final  result. 

The  operations  needed  in  consequence  of  the  above 
principles  are  therefore  the  following. 

1 .  Write  the  given  quantities  in  such  an  order ^ 
that  the  parts  containing  the  unknown  quantity  stand 
all  on  one  side  of  the  sign  of  equality ^  and  those  hav- 
ing none  but  known  qua7itities  on  the  other  side, 

2.  Arrange  it  so  :  that  the  square  of  the  unknown 
quantity  multiplies  at  once  all  the  quantities  which  it 
has  to  multiply,  and  do  the  same  with  the  quantities 
that  multiply  the  unknown  quantity  simply. 

3.  Disengage  the  square  of  the  unknown  quantity 
of  all  its  multipliers,  either  whole  or  fractional,  by 
dividing  every  term  of  the  equation  by  them. 

4.  Make  the  square  of  the  half  of  the  factors  which 
multiply  the  unknown  quantity  in  the  simple  form, 
and  add  this  square  to  both  sides. 

5.  Extract  the  sqvxire  root  of  tJiat  side  of  the  equa- 
tion which  has  no  icnknown  quantity,  and  write  on 
the  side  of  the  unknown  quantity  the  root  of  this  un- 
known quantity  and  of  the  square  added. 

6.  Subtract  the  part  added  to  the  side  of  the  un- 
known quantity  from  the  square  root  of  the  determin- 
ed number  of  the  other  side. 

7.  The  result  will  be  the  value  of  the  unknown 
quantity  sought. 

These  general  principles  will  include  all  cases- 
that  may  occur. 

§  100.  For  the  cube,  or  the  product  of  three  equal 
factors,  which  corresponds  in  nature  to  the  solid, 
we  have  to  multiply  the  product,  which  has  been 
obtained  for  the  square,  once  more  by  the  first  quan- 
tity;  in  order  to  show  what  different  parts  it  is 
composed  of,  the  aboTc  mode  of  separating  the  fac 


I 


S(tUJLR£  AND  CUBE  BOOTS*  161 

tors  is  to  be  preserved,  because  it  will  show  how  the 
products  are  to  be  made  in  the  extraction  of  the 
cube  root.  For  this  purpose  the  same  example^ 
which  has  served  before,  will  again  be  made  use  of. 
We  have  obtained  in  ^  98,  by  14  X  14,  or  142. 
the  result 

lOX  104-2  X  10X4+4X4    which  being  multiplied 
by  10x4        gives 

10X10x10+2X10x4x10+4X4x10 

+    4X10x10     +2X4X4X10+4X4>r4 

10X10xlO+3X10X10x4  +  3x  10X4X4  +  4X4x4 
=  103+3x10    X4  +  3X10X42+43   =  2744  =  14^ 

It  will  be  observed,  that  this  product  is  composed 
of  the  cube  of  10,*  three  times  the  square  of  10 
into  4 ;  three  times  the  product  of  10  into  the 
square  of  4  ;  and  the  cube  of  4.  Or,  generally,  the 
cube  of  the  first  part,  and  three  times  the  product 
of  the  square  of  the  first  part  into  the  second  ;  then 
three  times  the  product  of  the  first  into  the  square 
of  the  second  part ;  and  lastly,  the  cube  of  the  second 
part. 

Tliese  products  are  therefore  to  be  formed  out  of 
the  parts  of  a  cube  the  root  of  which  it  is  intended  to 
extract. 

It  will  again  be  observed  here,  that,  with  refer- 
ence to  the  subdivision  of  the  cube  in  the  order  of 
our  decimal  system,  the  second  term  will  be  the 
largest  after  the  first,  as  it  contains  the  double 
-square  of  the  first,  as  the  largest  factor  which  may 
occur  after  the  cube  of  the  first;  it  forms  therefore 
the  leading  part,  or  factor,  to  find  the  second  part, 
as  in  the  extraction  of  the  square  root. 

It  will  also  appear,  that,  as  we  had  to  divide 
off  the  number  into  pairs  of  figures  in  the  square, 
here  it  will  be  necessary  to  divide  off  the  number 
14* 


i65  StttTAEE  AND  CUBE  ROOTS^. 

every  three  figures,  from  the  right  hand  side  towards 
the  left,  because  the  product  of  a  number  of  two 
figures  into  one  of  one  figure  may  give  three  figures 
in  the  result. 

With  these  results,  and  the  principles  which  arise 
from  them,  for  the  converse  operation,  that  is,  the 
extraction  of  the  cube  root,  we  shall  be  able  to  exe- 
cute this  operation  properly. 

1st  Example.  The  above  resulting  number,  2744,  being 
given,  to  extract  the  cube  root,  which  is  indicated  thus  : 


First  cubic  root  taking  oflf  1  ^ 


-Remainder       1 

Divisor  =  3x10x10  = 

First  term  =300x4=       1 

Second  term  =3X10X4X4  = 
Third  term  =4X4x4  = 

Sum  of  the  three  terms  =       1 

Subtracted  from  the  remainder  == 


744      =  14 


744 


300    quot— 4 

200) 

480) 

64) 

744 


000 


The  only  number  which  cubed  will  not  exceed  2  is  1 , 
taking  away  this  cube  gives  the  remainder  1744  ;  forming 
the  triple  product  of  the  10^  =  300  ;  this  in  common  di- 
vision would  go  6  times  in  1744  ;  but  there  must  here 
be  room  for  the  subtraction  of  the  products  indicated 
above,  and  it  will  be  found  that  only  4  will  admit  that  : 
thereby  we  form  the  3  terms  placed  under,  as  indicated  j 
the  sum  of  which  is  equal  to  the  former  remainder,  and 
subtracted  leaves  0,  giving  14  the  exact  cube  root  of 
5744. 

2d  Example.  Extract  the  cube  root  of  994011992,  or 
execute 


V 

mp>        First  remainder 

Dinsor  =  3x9002  = 

3X900^X90  = 

3x900X90*  = 

903  -^ 

Sum  of  factors  =» 
Second  remainder  c= 
Divisor  =  3  X(990)2  = 

3x9903X8  = 

3X990X82  = 

83  = 

Sum  of  factors  = 

Third  remainder  = 


430 


Si^rABB  AKB  CUBB  BOOTS*  1<>5 

11992  =  9004-904-8  =  998 
000000 
On|992 

000       quotient  =  90 

000) 
000  > 

ooo) 

000 
992 
300      quotient  =  8 


»,994|0 
9003  =     729 


265 

sis 


21 

241 
23 


IP 


23 


700 
870 

729 


299 


712 


940 


522 
190 


23,712 

'ool'oo 


992 
~00 


This  gives  a  root  of  three  places  of  figures,  as  indicated 
by  the  partition.  The  nearest  cube  root  of  the  first 
division  of  the  numbers  on  the  left  being  9,  which  in 
the  third  place  is  equivalent  to  900,  the  cube  being 
made  and  subtracted,  leaves  the  first  remainder  ;  the 
triple  product  of  the  square  ©f  it,  taken  as  a  divisor^ 
shows  90  as  quotient,  for  the  root.  The  products  are  now 
formed  as  indicated  ;  their  sum  being  subtracted  from  the 
first  remninder,  leaves  the  second  remainder,  upon 
which  the  same  process  takes  place  as  before,  taking  the 
whole  of  the  root  found  as  the  first  term  ;  and  the  sum  of 
the  products  being  equal  to  the  last  remainder,  the 
Dumber  given  proves  an  exact  cube  of  the  number  ob- 
tained as  root. 

3d  Example.  If  the  number  is  no  exact  cube,  we  may 
extract  the  approximate  root  in  decimal  fractions,  as  well 
as  in  the  square  root ;  the  number  of  O's  to  be  added 
each  time  must  of  course  be  three,  and  the  products  are 
formed  as  required  in  the  former  example ;  the  pro- 
cess will  go  on,  in  other  respects,  as  has  been  seen  in  the 
square  root.     To  make  this  strikingly  apparent,  we  will 


hex'e  execute 


3,2  . 


thus: 


164 


SClUAllE  AND  CUBE  ROOTS. 


Remainder 

Divisor  SXIO^ 

3X102x2 

3x10x22 

23 

Sum  of  factors 
.  First  remainder 
Divisor  3X1202 

3X1203X5 

3X120X62 

63 

Sum  of  factors 

Second  remainder 

Divisor  3X1262 

3x12602x9 

3X1260X92 

93 

Sum  of  factors 

Third  remainder 

Divisor  3X126902 

3x126902X9 

3X12690x92 

93 

Sum  of  factors 
Fourth  remainder 


l,2699  +  &c. 


000 
000 
300 

600*1 
120  I 

728 

272  000 
43 


216 
9 


226 


46 
4 

42 

42' 
~4 


adding  three  O^s 


200 

000 
000 


126 


876 


68^ 


187 
303 


491 
383 


476 


279 
3 


282 


iOO 


000   adding  three  G's 

500 

500) 
750) 
729^ 

979 

000    adding  3  O'ir 

300 

700) 
370  > 

729) 

799 


021 


624 


718 
069 


778 


242 


201 


Adding  3  0's,it  would  be  continued  as  before. 

The  place  of  the  decimal  mark  is  evidently  again  de- 
termined by  the  usual  principle,  namely  :  where  it  be- 
comes necessary  to  add  O's  to  continue  the  opera- 
tion. 

§  101.  We  here  see  again,  that  the  principles  de- 
duced may  lead  to  the  solution  of  equations  of  the 
third  degree,  as  this  is  called  in  higher  calculations. 


p:B;OGR£SSIOfirS   OB   SEBIB9.  165 

or  to  determine  a  quantity  which  appears  as  form- 
ed of  three  equal  factors  multiplied  into  each  other  ; 
but  it  is  not  the  province  of  arithmetic  to  go  into  this 
inquiry ;  because  it  requires  operations,  and  pro- 
duces cases,  which  are  reserved  to  be  solved  only 
in  universal  arithmetic,  or  algebra. 

It  is  evidently  possible  to  produce  the  involutions 
of  higher  degrees  in  the  same  manner  that  has  here 
been  shown  for  the  square  and  the  cube ;  but  the 
evolution  presents  increasing  difficulties  as  we  pro- 
ceed, the  possible  combinations  of  different  factors 
to  the  same  ultimate  result  being  evidently  always 
more  numerous,  and  thei^fore,  also,  the  possible 
roots.  Even  in  algebra  there  is  not  yet  a  general 
method  found  to  solve  such  questions,  and  it  steps 
entirely  out  of  the  limits  of  arithmetic  to  treat  any 
thing  relating  to  this  subject. 


► 


CHAPTER  II. 

Of  Progressions  or  Series, 


§  102.  In  mentioning  (§  84  and  89)  continued 
proportions,  and  the  progressions  or  series  which 
result  from  their  continuance,  we  referred  to  a 
future  extension  of  the  subject  to  the  progressions 
or  series,  which  are  intended  as  the  subject  of  the 
present  chapter. 

According  as  the  continued  proportion  is  either 
an  arithmetical  or  a  geometrical  proportion,  we  ob- 
tain by  its  extension  to  a  greater  number  of  qanti- 
ties  :  either  an  arithmetical  or  a  geometrical  progres- 
sion, or  series;  each  of  which  has  peculiar  laws ;  we 
shall  here  begin  with  the  first. 

$  103.  A  series  of  numbers  which  progresses  in- 
creasing, or  decreasing,  by  the  same  constant  differ^ 


16^  FROGKESSIOXS   OR    SERIES, 

ence,  forms  a  continued  arithmetical  proportion^  or 
an  arithmetical  series. 

This  principle  is  therefore  the  element  of  all  in- 
vestigation in  relation  to  the  properties  of  this  kind  of 
series;  according  to  it  we  shall  be  able  to  write  all 
tlie  terms  successively,  and  therefore  obtain  the  la\s 
of  the  mutual  dependance  of  all  the  quantities  con- 
cerned in  it ;  such  a  series  (which  we  will  caii  equal 
to  SJ  will,  for  instance,  be  the  following  : 

5  =  2-}-(2+3)-}-(2+2  X3)4-(2-f  3  X3)-i-(2-4-4X3)H-(2-f  5X3)+ &c. 

In  the  writing  of  these  series  the  terms  are  joined 
by  the  sign  +»  which  ma^^  equally  serve  to  express 
tlie  arithmetic  proportion,  as  I  stated  at  first,  and 
the  constant  equality  of  the  difference  will  become 
equally  apparent  by  the  subtraction  of  each  term 
from  its  immediately  subsequent  term^  which  gives 
liere  the  constant  difference,  3. 

Considering  the  successive  dependance  of  these 
terms  upon  each  other,  and  comparing  their  value 
in  relation  to  their  distance  from  the  first  term,  we 
observe  that  the  constant  difference  makes  its  first 
appearance  in  the  second  term,  and  being  afterwards 
found  added  in  each  subsequent  term,  it  will  in 
any  term  whatever  be  one  less  than  the  number 
of  terms  indicates,  whether  the  series  be  increasing 
or  decreasing.  Thus  we  find  it  here  in  the  sixth 
term  added  five  times  to  the  first  term.  This 
gives  us  the  principle  by  which  to  determine  any 
term,  when  the  first  term  and  the  constant  difference 
arc  given. 

It  will  be  of  the  greatest  advantage  in  the  exten- 
sion of  arithmetic  in  this  state  of  foi  wardncss, 
to  apply  the  use  of  letters  to  denote  certain  quan- 
tities, until  tliey  are  determined,  that  we  may  ex- 
press our  ideas  clearly,  fully,  and  briefly,  by 
applying  to  them  the  signs  of  arithmetic  which 
hare  been  taught  in  the  beginning.     We  will  there- 


I 


PROGKESSIONS    OB   SERIES.  IST 


fore  generally  denote  the  quantities  concerned  in  onr 
present  investigation  by  proper  letters ;  thus : 
Xict  the  first  term  be  designated  by,  or  =  a 
"       constant  difference  by  =  d 

"       number  of  terms  of  the  series      =  n 
"       sum  of  the  series  =  S 

Thus  we  shall  be  able  to  express  the  property, 
which  we  have  just  found,  of  the  value  of  any  term, 
which  we  denote  by  ii,  by 

term  (n)  =  a  -{-  (n—  \)  d 
And  the  whole  series  extended  to  the  term  n,  would 
be  written  thus,  (omitting  the  intermediate  terms  :) 

1st      2d  (n-l)st  nth 

.S  =  a4-(a  +  t/)  ....  (rt+(7i-2)d)+(a-f(n-l)d)+&c. 

Considering  the  Tith  term,  it  is  evident  that  if,  of  tlie 
three  quantities  concerned  in  it,  and  the  whole  value 
of  the  term  itself,  any  three  are  given,  the  fourth 
may  be  determined  from  them,  just  as  we  determin- 
ed the  fourth  term  in  a  geometrical  proportion,  not- 
withstanding tliat  the  law  of  their  mutual  depend- 
ance  is  very  different. 

Example.  In  the  above  series  we  had  o  =  2  ;  d  =  3  ; 
let  n  denote  the  sixth  term.  We  shall,  by  putting  the 
values  of  the  letters  in  their  places,  and  performing  the 
operations  indicated,  obtain  the  following  : 

Value  of  the  6th  term  =  2  +  5x3  =   17 

In  a  similar  manner  any  other  term  would  be  obtained, 
as  : 

The  21st  term  =  2  +  20  X  3  =  62     and  so  on. 

If  we  had  62  as  the  value  of  the  term  given,  and  the 
6rst  term,  together  with  the  constant  difference,  we 
would  evidently  obtain  the  number  corresponding  to  the 
term,  by  subtracting  the  first  term  from  the  sum,  and  di- 
viding the  remainder  by  the  difference,  then  adding  a 
imit  to  the  quotient ;  thus  : 


16$  FBOGRESSIOXS    611   SEIIIE9. 

62-2  =  60 ;  then  V  =  20.  Adding  1  gives  for- 
n  «=  21. 

In  like  manner  any  other  part  can  be  found,  by  revers- 
ing the  operations  accordingly. 

§  104.  The  most  frequent  use  of  these  series,  and 
therefore  the  principal  object  of  inquiry,  is  the  de- 
termination of  their  sum  by  means  of  the  three  other 
quantities  concerned  in  it.  The  principle  of  this 
determination  is  deduced  from  the  nature  of  the  se- 
ries, in  the  following  manner. 

As  we  found  in  arithmetic  proportion  that  the  sum 
of  the  extremes  is  equal  to  the  sum  of  the  means,  so  it 
is  evident  that  here  the  sum  of  the  extremes  is  equal 
to  the  sum  of  any  two  terms  equally  distant  from 
them,  for  the  sum  of  every  such  pair  of  terms  must 
contain  the  first  term  twice,  and  the  constant  diifer- 
ence  an  equal  number  of  times,  because  these  in- 
crease in  numbers  equally  from  the  beginning 
onward,  as  they  decrease  from  the  end  backward. 

In  the  above  series  we  obtain  : 

By  the  first  and  last  or  6th  term  : 
2  +  2  +  5X3   =    19 
By  the  second  and  last  but  one,  or  fifth  term : 

2  +  3  +  2  +  4X3   =    19 
By  the  third  and  fourth  term  : 

2  +  2X3  +  2  +  3X3   =    19 

And  generally,  by  the  first  and  nth  term,  we 
would  obtain,  adopting  the  expressions  above  used, 
the  general  value  of  any  pair  of  terms  : 

a  -{-  a  -{-  {n  —  1)  d 

Summing  up  all  these  pairs  of  terms,  we  would  of 
course  obtain  the  sum  of  the  whole  series.  But 
there  are  as  many  pairs  of  terms  as  the  number  of 
terms  divided  by  2 ;  therefore  we  may  obtain  the 
value  of  the  whole  series  at  once,  by  multiplying 


(  PKOGRESSIOX   OB    SERIES.  169 

e  value  found  above  by  half  the  number  of  terms  i 
that  is,  in  the  above  numbers  : 

(2  -f  2  +  5  X  3)  f   =   57 
And  in  the  general  expression  in  letters,  or,  as  this 
is  usually  called,  equation  : 
n 
^  =  —  (2  a  -I-  («  -  1)  d) 
2 
In  this   general  expression  again  there  are  only 
four  quantities   concerned,  three  of  which   being 
given  the  fourth  is  determined,  by  making  such 
operations  upon  the  above  equation  as  will  bring  the 
quantity  to  be  determined  alone  on  one  side  of  the 
sign  of  equality,  as  in  this  case  the  S, 

§  105.  To  determine  any  quantity  in  any  way 
involved  in  such  an  expression  as  the  above,  which 
in  general  arithmetic  is  called  an  equation,  the  same 
principle  is  made  use  of  as  has  been  shown  in  pro- 
portion, namely,  that  all  such  mutations  are  allowed 
as  do  not  change  the  principle,  that  after  the  change 
made,  the  quantities  on  each  side  of  the  sign  of 
equality  are  again  equal.  This  leads  directly  to 
the  consequence,  that  we  are  allowed  to  perform 
any  operation  of  arithmetic  we  may  wish  upon 
such  an  equation,  provided  we  do  the  same  on 
both  sides. 

As  we  have  seen  above,  that  the  operations,  com- 
monly called  rules  of  arithmetic,  ai'e  of  such  a  na- 
ture, that  two  are  ahvays  opposite  to  eac  h  other, 
that  is  to  say,  the  one  will  alv.  ays  evolve  what  the 
other  has  involved,  or  disengage  what  the  other  has 
engaged,  we  shall  naturally  in  an  operation  such  as 
is  proposed  always  perform  upon  such  an  equation 
success  ively  all  the  operations  which  will  disen- 
gage the  quantity  from  all  others,  until  it  ulti- 
mately be  found  alone  on  one  side  of  the  sign  of 
equality. 

15 


iro  pro6ressio:n^  or  series. 

We  will  therefore  now  apply  these  principles  to 
the  equation  before  us,  to  obtain  successively  expres- 
sions or  equations  for  each  of  the  quantities  by 
means  of  all  the  others. 

1st  Problem,  To  find  the  first  term  of  the  series, 
knowing  all  the  other  parts,  we  would  proceed  thus  : 
Taking  the  original  equation 

n 
8  =  (2  a  4-  (w  -  1)  <i)  — 

n 
we  will  divide  on  each  side  by  — ;  which  will  dis- 

2 
engage  this  multiplication,  and  give  : 

=  2a  -f  {n  -  I)  d 

n 
Then,  in  order  to  disengage  the  addition  on  the 
right   hand  side,    we   will    subtract  on  each  side 
what  is  added  there,  to  the  part  containing  the  first 
term^  this  changes  the  equation  thus: 

{n  —  1)  d  =  2  a 

n 
The  a,  or  first  term,  will  now  be  alone,  and  there- 
fore be    determined,   if  we   divide   on    each   side 
by  2,   this  gives  ultimately  : 

S  d 

(n  -  1)  —  =  a 

n  2 

This,  expressed  in  words,  which  is  in  fact  a  less 
convenient  way  than  the  above  exm-ession,  which 
speaks  to  the  eye  at  once,  would  be  thus :  the  first 
term  is  equal  to  the  difference  between  tiie  sum  of 
the  terms  divided  by  tlie  number  of  terms,  and  the 
product  of  lialf  the  common  difference  into  the  num- 
ber of  terms  less  one. 


PROGRESSION   OR    SERIES.  171 

Suppose  we  had  the  sum  of  the  series:  S  =  164 

"              "             common  difference :  d  =       5 

*'             "              number  of  terms:  n  =       8 

the  above    expression   would   present  us  the  following*. 

result : 

1G4         7X5          164         140  24 


8  2  8  8  8 

Qd  Problem,    To  find  the  difference,   we  would 
mutate  the  equation  after  the  first  step  thus  : 
■^   Having 

IK  ^^ 

P 

IF 'we  subtract  2  a  on  each  side,  which  gives  : 

2  a  =  {n—  1)  d 


f» 


nd  we  divide  by  ?i  —  1  on  both  sides,  which  gives 
the  result : 

28  2a 

n  (n  —  1)       n  —  1 

This  expression  can  be  made  more  convenient  for 
calculation,  by  subtracting  the  fractions  after  re- 
duction to  a  common  denominator.  Thus  it  be- 
comes : 

2  S  -  2na 

d  = 

n  {n  —  1) 

And  by  making  2  a  common  multiplier  to  both 
terms  of  the  numerator : 

2  {S-na) 
n  (n  —  1) 


17'2  PROGRESSION    OR    SERIES. 

Assuming  for  the  letters  the  values  given  to  them 
above,  we  obtain  i 

2  (164  -  8X3)  2  X  140  280 

8X7  56  66 

3d  Problem.  To  determine  any  term  of  the  series, 
liaving  given  the  first  term  and  the  common  differ- 
once. 

From  the  nature  of  the  series  we  have  seen,  that 
each  term  after  the  first  has  always  the  common 
difference  added  to  it,  in  order  to  form  the  subse- 
quent one ;  therefore  each  term  is  determined  by 
adding  to  the  first  term  the  common  difference  as 
many  times  as  the  number  of  the  term  required  indi- 
cates, less  one,  thus : 

Having  the  first  term  =  6;  the  common  difference  4j 
the   17th  term  will  be  =  6  +  16  X  4  =  69. 

4th  Problem.  Any  two  terms,  the  first  being  one 
of  them,  and  the  common  difference  being  given,  to 
find  the  number  of  terms. 

When  the  first  term  is  subtracted  from  the  other 
term  given,  we  have  the  product  of  the  common 
difference  into  the  number  of  terms  less  one  as  re  ^ 
mainder ;  dividing  this  therefore  by  the  Ci)mmon 
difference,  we  have  the  number  of  the  term,  when 
we  add  one  to  this  quotient;  as  for  example: 

The  first  term  being  5  ;  the  other  term  given  69  ;  the 
common  difference  4 ; 

Subtracting  the  first  term  gives    69  —  6  =  64  ; 
Dividing  this  by  4,  we  obtain  =  ^6  ;  to  which  adding 
1,  gives  the  number  of  the  term  =17. 

5th  Problem.  To  find  the  distance  which  two 
terms  in  an  arithmetical  series  are  from  each  other, 
the  common  difference  being  given : 

If  we  subtract  the  two  terms  from  each  other,  we 
evidently  have  for  the  remainder  the  product  of  tho 


I 


I 


PROGRESSION   OB    SERIES.  173 

onimon  difference  into  the  difference  between  the 
terms ;  therefore,  when  we  divide  this  remainder  by 
the  common  difference,  we  obtain  the  number  ex- 
pressing the  distance  of  the  terms ;  as  for  example : 

Having  the  two  terms  69  and  92,  and  the  common 
difference  4,  we  obtain  97  —  69  =  28  ;  dividing  by  4, 
the  distance  of  the  terms  becomes  3=  7. 

These  problems  may  evidently  be  varied  in  dif- 
ferent ways ;  and  i  now  allow  myself  the  supposi- 
tion that  the  scholar  will  be  able  to  do  it  by  himself, 
as  he  may  wish  or  need  it. 

6th  Problem,  The  sum  of  the  series,  the  first  terra, 
and  the  constant  difference,  being  given,  to  find  the 
number  of  terms. 

This  solution  will  lead  us  into  a  quadratic  equa- 
tion, the  principles  of  which  have  been  explained 
above,  with  the  express  view  to  their  application  in 
this  chapter.  It  is  proper  to  treat  it  in  the  general 
form ;  we  shall  therefore  take  the  first  formula,  or 
equation,  for  the  sum  of  the  whole  series,  and  from 
it  solve  the  value  of  n,  by  the  following  successive 
steps : 

n 

Original  equation,      S  =  —  (2  a  -|-  (71  ~  1)  d) 

2  ' 

Multiplying  all  by  2  : 

QS  =  n  (2  a  4.  (w-  1)  d) 

Executing  the  multii)Iication  by  11,  indicated, 
and  also  by  d,  in  its  place  : 

QS  =  2  a?i  -h  dn^  ~  nd 
Arranging  the  parts  on  the  right  by  the  powers  of 
n,  and  making  n  the  common  factor  to  its  multipli- 
rrs  in  the  two  terms : 

2  S  =  dn^  4-  (2  ft  -  d)  w 
Dividing  by  d  to  make  the  n^  free  of  factors  : 
15* 


174  PROGRESSION   OE   SEKIES, 

Q  8  Qa- d 

, =  n^  H .  .  n 

d  d 

2a^d 

The evidently  represents  here  tlie  double 

d 

of  the  second  term,  which  we  found  above  in  a  qua- 
dratic equation ;  taking  then  the  half  of  it,  squaring 
it,  and  adding  it  on  both  sides,  gives  : 

2S  /2a-d\^  Qa-d  /9.a~d\^ 
—  -{-  I I    =  n^  + 71+1 I 

d  \    Qd  /  d  \  2d   ^ 

The  square  root  can  be  extracted  on  the  right  side, 
it  being  an  exact  square ;  there  being  on  one  side  none 
but  known  quantities,  that  are  equal  to  the  sum  of  a 
known  quantity  and  tlie  quantity  sought,  this  latter 
will  ultimately  be  obtained  by  a  simple  subtraction. 
These  opertions,  expressed  by  known  signs,  give : 

'2.S        /2a-d\'i\  2a  ^d 


X2,8        /2a-d]^'i\ 


d  \    2d     '  '^  2d 

Which  we  will  now,  by  way  of  explanation  in  num- 
bers, apply  to  the  numerical  series  supposed  m  the  first 
problem  above,  by  placing  for  each  letter  (except  the 
unknown,  »)  its  value. 

,/2Xl64         /2x3-6v,^  2X3-5 

/( +  ( )-  )  =  r*  + 

V\      5  V    2x5    /    /  2X5 

Bringing  the  parts  of  which  the  root  is  to  be  extracted 
under  one  single  number,  by  the  following  operations 
successively  : 

328     1     328x20+1    6561 

j =r =  «  65j6l 

5     100       100       100 


IHF  FSOGRESSIOX   OB    SERIKS.  175 

ihe  above  will  give  us  : 

-v/  65,  61   =  n  +  tV  =  S»  1 
and  n  =  8,  1  —  0,  1   =  8 

§  106.  We  have  seen  in  §  88,  that  the  continuance 
of  a  geometrical  proportion  produces  a  series  of 
quantities  of  which  each  subsequent  is  a  prodtict  of 
the  preceding  one  by  a  canstant  factor,  either  whole 
or  fractional :  the  fii  st  case  producing  an  increas- 
ing, and  the  second  a  decreasing  geometric  series 
for  progression^ J  which  is  therefore  the  constant 
ratio  between  the  terms. 

The  principles  of  the  geometric  series  are  appli- 
cable in  all  questions  that  relate  to  compound  inte- 
rest, annuities,  and  the  like ;  their  principles  will 
here  be  investigated  in  a  manner  similar  to  that 
used  for  the  aritlunetical  series,  but  upon  the  prin- 
ples  of  the  geometric  propoHion,  of  which  it  is  the 
continuance.  We  will  for  that  purpose  proceed  by 
the  example  of  the  following  series;  the  sum  of 
which  we  again  call  S,  to  have  a  point  of  compari- 
son ;  the  terras  are  therefore  also  added,  or  joined 
by  the  sign  (-}-). 
S^  3+ox  34-58  x:3-|-5»  X3+54  X  34-55  X3-I-56  x  3  &c. 

The  law  of  continued  geometric  proportion,  that  the 
product  of  the  tw  o  extremes  is  equal  to  the  product 
of  the  mean  term  into  itself,  evidently  holds  good 
here,  and  we  have,  for  instance,  by  the  product  of 
riic  first  and  third  term,  compared  with  the  second., 
the  following  results : 

3X52X3   =   5X3X5X3 
or  2£5   =   225 

And  by  the  same  process  upon  the  last  term  and  the 
second  before  the  last,  compared  with  the  one  before 
the  last : 

5^X3X5'5XS   =    55X3X5^X3 
or         52197165625   =   2197165625 


176  PROGRESSION   OR   SERIES. 

In  both  cases  results  evidently  identical  are  ob- 
tained. 

Comparing  the  number  of  the  factors  of  the  con- 
stant ratio  in  each  term  with  the  number  of  this 
term,  we  find  again,  as  in  the  arithmetical  series  : 
that,  as  this  factor  ap])ears  of  course  for  the  first 
time  in  the  second  term,  each  term  will  contain  one 
factor  less  than  the  number  of  tliis  term ;  thus  the 
second  term  has  one  factor,  the  third  two,  the 
seventh  (as  above)  six ;  and  in  general  the  nth  term 
will  have  n  —  I  factors,  exactly  in  a  similar  man- 
ner as  found  in  arithmetical  series.  This  con- 
sideration enables  us  to  determine  any  term  of  the 
series,  for  the  7ith  term  of  the  series  above  will  be 
=  3X5^""^ '5  and  if  we  again  adopt  general  deno- 
minations as  in  ainthmetical  series,  by  calling 
the  first  term  =  a 
the  constant  ratio  =  r 
we  would  write  the  above  expression  of  the  nth. 
term  =a  .  r*""^^ ,  that  is  :  the  71th  term  is  equal  to 
the  product  of  the  first  term  into  the  common  ratio 
elevated  to  a  power  one  unit  less  than  this  number 
of  the  term.  We  may  therefore  again  determine 
any  one  of  these  four  quantities  when  we  have  the 
three  others  given. 

^107.  From  t\\G  principles  of  continued  geometric 
2)7'opoi'tion  a  formula,  or  equation,  is  now  to  be  de- 
duced, expressing  the  sum  of  a  geometric  series  in 
general  terms.  We  have  seen  among  the  mutations 
of  the  geometric  proportion  :  that  the  sum  of  the  two 
terms  of  eac  h  ratio  may  be  compared  with  either  its 
antecedent  or  its  consequent ;  this,  applied  to  conti- 
nued pi'oportion,  where  the  middle  terms  are  equal, 
produces  the  following ;  applied  as  example  to  the 
first  three  terms  of  the  above  series,  namely  : 

3:3X5   =  3X5:3X52 
whence,  by  addition : 

3-f3X5:5  =  SX5i-QX5^  :SX5 


PROGRESSIOJf   OK    SERIES.  177 

This  change  might  evidently  be  carried  on  through 
the  whole  extent  of  the  series,  and  we  might  there- 
fore have  the  sum  of  all  the  antecedents  in  the  first 
antecedent  above,  and  the  sum  of  all  the  consequents 
in  the  second  antecedent ;  or  by  expressing  the  sum 
of  all  the  antecedents  by  the  sum  of  tlie  whole  series 
less  the  last  term,  and  the  sum  of  all  the  conse- 
quents by  the  sum  of  the  series  less  the  first  term, 
we  will  have  a  general  proportion  resulting,  expres- 
sed in  the  letters  adopted  above,  and  for  a  series  of 
n  terms;  viz: 

Sum  of  an- 1     ^   ..  f  sum  of  con-")     ^^. 

tecedents    j  *•  1«**^^«^  =  1   sequents    j  ''  ^^^''''''' 

S-ar^""'  :  o  =  S  -  a  :  ar 

or,  by  mutating  the  middle  terms  : 
IK  ^— .  ar^"'^   :  8  -  a  =^  a  :  ar 

and  by  subtraction : 

a  —  ar  '"^^ :  S  —  a  =  a  -  ar  i  ar 
Dividing  the  antecedents  by  a: 

1  -r^-^-i'  :  8  -  a  =   \  -  r  :  ar 
Multiplying  the  antecedents  by  r : 

r  —  r"  :  8  -  a  =  r  (1  -  r)  :  ar  =   1  ~  r  :  « 
Exchanging  the  mean  terms  : 

r  -  r"  :  1  —  r  =   8      a  :  a 

Sum  of  antecedents  and  consequents  compared  with 
the  consequents  : 

r-r"-fl-r:  l—r  =  8  -^  a  +  a  :  a 
or         I  -  r"  :  I  -^r  =  8  :  a 
I  which  gives : 

a  (    -  r°)         a  -  ar"         ar"  —  a 

l—r  l—r  l  —  r 

The  better  to  impress  this  operation,  and  its  different 


178  PROGRESSIOJf   OR   SERIES. 

steps,  I  will  repeat  it  here  in  the  numbers  of  the  above 
series,  which  will  enable  us  to  make  the  full  comparison 
of  its  general  result  with  any  individual  case  that  may 
occur.  The  series  chosen  gives  the  following  numbers 
in  the  first  proportion,  under  the  supp»€ition  of  the  num- 
ber of  terms  n  being  7. 


S— 3  X  6 

7  1  1 

:  3 

= 

S- 

-3:3X6 

s 

—  3  X  58 

:  S- 

-3 

= 

3  : 

3x5 

z 

-3  X  5«  ; 

:  S- 

-3 

= 

3 

—  3x6 

:  3  x  5 

1  -5«  : 

:  S. 

-.3 

= 

l- 

-6  :  3x 

5 

5  — 67  : 

S- 

-3 

= 

6- 
1  - 

-6X5: 
-6  :  3 

3X6 

6  — 67  : 

1  - 

-6 

= 

S- 

-3  :  3 

b  -  5'' 

+  1-5: 

:  1 

-6 

= 

S- 

-3  +  3  : 

3 

1  —  67  : 

;  1- 

-  6 

= 

S: 

3 

S     =3 

I  — 

67) 

3- 

-3X67 

1  —6 

234372 


=  68593 

4 

Remark.  I  here  permitted  the  quantity  to  be  sub- 
tracted to  be  the  greater,  both  in  the  numerator  and  in 
the  denominator  ;  this,  though  apparently  a  contradiction, 
is  compensating  on  the  same  ground  as  has  been  shown 
above  :  that  the  objects  themselves  disappear  in  a  rule 
of  three,  when  they  appear  equally,  both  in  numerator 
and  in  denominator  ;  the  result  here  is  therefore  equally 
positive.  The  signs  of  addition  or  subtraction,  that  is, 
+  ,  and  — ,  compensate  as  equal  quantities  in  numerator 
and  in  denominator,  exactly  like  the  ouanlities  them- 
selves. It  will  easily  be  seen,  that  if  the  series  had 
been  a  decreasing  one,  the  case  would  have  been  the 
reverse  ;  the  ratio  being  in  that  case  a  fraction,  the  nu- 
merator and  the  denominator  would  both  have  presented 
positive  numbers,  that  is,  the  subtracting  quantities,  be- 
ing fractions,  would  both  be  smaller  than  the  nnd. 

The  above  expression  for  the  value  of  the  sum  of 


PROGRESSION   OR   SERIES.  179 

a  geometric  progression  is  therefore  the  rule  (to  ex- 
press it  in  the  common  language  of  arithmetic)  by 
which  this  sum  is  to  be  calculated.  It  can  be  stated 
Tery  simply  thus : 

Take  the  difference  between  unity  and  the  constant 
ratio  elevated  to  the  power  indicated  by  the  number  of 
terms,  divide  this  by  the  difference  between  unity  and 
the  constant  ratio,  and  multiply  the  quotient  by  the 
first  term. 

This  rule  is  evidently  adapted  both  to  increasing 
and  to  decreasing  geometrical  progressions. 

§  108.  The  foregoing  expression,  or  formula, 
again  presents  us  four  quantities  mutually  depend- 
ing upon  each  other,  in  the  manner  expressed  by  it ; 
we  may  therefore  conclude  :  that  any  three  of  tliem 
given  determine  the  fourth ',  which  might  form  as 
many  distinct  problems,  as  shown  in  the  arithmetic 
series ;  we  will  here  only  show  how  to  find  the  first 
term,  the  other  parts  being  given. 

The  last  step  of  tlie  reduction  of  tlie  proportion 
evidently  gives : 

1-r 

l~r" 

or,  in  words :  Divide  the  difference  between  unity 
and  the  constant  ratio,  by  the  dfference  between  unity 
and  the  ratio  elevated  to  the  power  indicated  by  the 
number  of  terms,  and  multiply  the  quotient  by  the  sum 
of  the  series. 

To  determine  the  constant  ratio,  or  the  number 
of  the  term,  when  the  other  parts  are  given,  re- 
quires more  extensive  deductions  and  calculation 
than  the  plan  of  these  elements  achnits  of;  the  first 
requires  the  solution  of  what  is  called  a  higher  equa- 
tion, and  the  second  the  use  of  logarithms,  wliich 
hoth  lie  beyond  our  present  limits. 


180  COMPO¥N»    INTEBESi:'. 

CHAPTER  III. 

Of  Compmind  Interest. — Idea  of  Annuities* 

§  109.  We  have  seen  in  its  proper  place,  that  the 
calculation  oi  simple  interest  was  a  simple  multipli- 
cation of  the  capital  by  the  decimal  fraction  repre- 
senting the  interest  per  hundred,  and  in  the  Com- 
pound Rule  of  Ti»ree  the  other  questions  have  been 
treated  which  relate  to  this  subject.  But,  as  well 
for  the  transacti«)ns  of  monied  institutions,  as  for 
various  other  calculations,  in  political  economy  and 
otherwise,  the  interest  after  the  year,  or  any  other 
term  agreed  upon,  is  considered  as  again  bearing 
interest,  and  thus  the  interest  increases  at  the  same 
rate  as  the  capital  itself.  This  introduces  of  course 
a  mode  of  calculation  completely  different,  and  par- 
taking of  the  nature  of  the  Progressions :  its  prin- 
ciples shall  here  be  treated  separately,  and  with 
the  addition  of  payments  at  determined  terms, 
as  the  interests  or  annual  payments,  called  annul- 
tieSf  of  which  it  may  be  proper  here  to  give  only  the 
first  principles,  without  going  into  the  details  which 
more  intricate  speculations  introduce  into  them,  as 
they  would  draw  us  out  of  our  prescribed  limits. 

We  shall  take  the  liberty  of  making  use  of  letters 
to  designate  the  quantities,  until  we  give  them 
actual  values,  by  way  of  example ;  in  order  to  give 
i.0  the  reasoning  that  general  form  which  it  is  so 
advantageous  to  introduce  in  the  higher  branches 
of  arithmetic.  Thus  we  will  call  the  capital  =  0, 
and  the  rate  of  the  per  centage  =  r :  and  proceed 
with  these  as  if  they  were  known  numbers,  indicat- 
ing the  operations  by  means  of  the  signs  which  we 
have  long  been  familiar  with. 

The  capital  having  been  one  year  at  interest^  it 


tAND   ANNtTITIES.  181 

ill  be  worth,  together  with  that  interest, 

C-f-rC  =  C(l  -f  r) 
(for  the  C  multiplies  the  unit  and  the  rate  per  cent, 
=  r.)  This  being  now  the  capital  on  interest  for 
the  second  year,  it  will  produce  an  interest  = 
C  (1  -f  r)  r ;  and  the  whole  value  of  the  capital  and 
interest  at  the  beginning  of  tlie  third  year  will  be 
the  sum  of  the  last  year's  capital  and  the  interest 
of  the  same,  namely  : 

C(l-f-r)  +  C(I-}-r)r  =  C(l+r)  (l+r)  =  C(l+ry 
(for  here  the  C  (1  -{-r)  is  again  a  multiplier  for  the 
unit  and  the  rate  per  cent.  =  r,  and  so  in  each 
following  year.)  Tiiis  capital,  at  the  same  interest, 
in  the  third  year  will  produce  an  interest  = 

C.r  (1  -f  r)2 
which  added  to  the  last  capital,  gives  at  the  begin- 
ning of  the  fourth  year  the  value  of 

C(l-f-r)^-fCr(l+r)«  =  C(H-r)a(l-hr)  =  C(l+r)=' 
This  is  therefore  the  law  of  the  increase  of  a  capi- 
tal  put  out  upon  compound  interest;    which  for 
any  number  of  years,  say  n,  would  give 

C(l  +  ry»=/Sr;     or. 
In  order  to  obtain  the  value  of  the  whole  capital  at  the 
end  of  the  last  year,  the  rate  of  interest  added  to  unity, 
raised  to  the  power  indicated  by  the  number  of  years 
elapsed,  is  to  be  multiplied  into  the  original  capital. 

To  show  the  same  operation  in  numbers,  let  us  sup- 
pose a  capital,  C=  7500,  at  the  rate  of  6  percent, 
compound  interest ;  this  (expressing  the  per  centage  in 
a  decimal  fraction)  evidently  gives  : 
The  first  year's  interest : 

7500  X  0, 06 
The  capital  at  the  end  of  the  first  year  : 
7500  -f  7500  X  0,  06 
which  will  be  more  easily  calculated  thus  ; 
7600  X  3,06 
16 


18£  COMPOUND   INTEKEST 

The  second  year's  interest  will  be  : 
7600  X  1,6X  0,06 
The  capital  at  the  end  of  the  second  year : 
7500X1,16  +  7500    Xl,06Xo,06 
or,  again  expressed  more  simply  : 

7500X1,06  X  1,06  =  7500(1,06)2 
It  will  progress  in  this  manner  every  year  by  the 
power  of  1,  06  ;  that  is,  the  original  capital  will  be  mul- 
tiplied  by  1,06  in  continued  multiplication  of  as  many 
factors  as  the  number  of  years  indicates  ;  for  instance, 
at  the  end  of  six  years  we  would  have  : 

$  7500  (1  X  6)6   =  7500  X  1,26247696 

§  110.  If  to  the  above  condition  of  compound 
interest  we  add  the  condition  of  annual  payments, 
we  hav«  the  idea  of  an  annuity^  when  these  pay- 
ments are  supposed  larger  than  the  interest,  (as  in 
that  case  the  whole  might  be  reduced  to  simple  inte- 
rest,) it  is  evident  that  they  must  eventually  consume 
the  capital  itself,  and  that  compound  interest  must 
also  be  allowed  upon  these  payments  as  well  as 
upon  the  capital ;  the  conditions  of  such  contracts 
are  therefore  varied,  and  grounded  upon  various 
contingencies,  and  principally  upon  a  combination 
of  chances,  particularly  the  probabilities  of  life,  into 
which  it  cannot  be  our  object  to  enter;  tlie  first 
principle  which  lies  at  their  root  is  all  that  is  in- 
tended to  be  shown  here.  The  diffei-ence  between 
the  capital  increased  at  compound  interest,  and 
the  payments  made,  at  any  time,  allowing  the  same 
rate  of  interest,  is  therefore  the  value  of  the  an- 
nuity at  that  time;  this  will  be  founded  upon  the 
following  investigation. 

We  shall  here  proceed  as  in  the  preceding  section, 
calling  the  annual  payment  =  p;  and  supposing 
them  to  begin  at  the  end  of  the  first  year,  it  will 
afterwards  be  easy  to  adapt  the  result  to  other  con- 
ditions of  payments,  beginning  at  a  later  periou 
Thus  we  have, 


AND    ANNUITIES.  183 

It  the  end  of  the  first  year,  the  amount  left 

=  r(l+r)-p 
it  the  end  of  the  second  year 

=  0(1  +ry  -p{l^r)^p 
At  the  end  of  the  third  year 

=  C{l-{-ry-p{l  -^ry  '-p{l-\-r)''P 
and  so  on  every  subsequent  year,  always  deducting 
from  the  original  capital  with  its  compound  interest 
at  the  time,  the  payments  made  with  their  interests, 
at  tlie  same  rate,  also  at  compound  interest. 

So  for  the  end  of  any  year,  generally  named  =  n, 
we  shall  have  for  the  amount  left,   called  a,  ex- 
pressed as  follows:* 
a  =  C(l  -\-rY-p{\  4-r)'»-i— ;?(!  +  r)°-2  until  —  p 

The  series  of  payments  with  their  interests  evi- 
dently form  a  decreasing  geometrical  series  with 
the  constant  ratio  =  (1  -f  r),  the  payment  =  p, 
being  the  first  term ;  we  can  therefore  place  its 
value  at  once  instead  of  the  series  according  to  the 
expression  found  in  §  107.  The  number  of  terms 
is  evidently  =  iu  because  the  payments  are  conti- 
nued until  the  term  p,  which  has  not  the  common 
ratio  in  it.     So  we  have  again 

l-(l-f  r)» 

a  =  C{1  +ry-p  X 

1       (1-r) 

=  C(I  -i-r)"-p  X 

r 

If  the  payments  were  to  commence  at  a  later  period 
than  the  beginning,  or  to  stop  after  a  certain  num- 

*  To  express  this  in  a  rule  would  be  useless  ;  we  will  rather  sub- 
sliiute,  by  way  of  example,  the  numbers  which  the  letters  repre- 
sent, and  join  the  result  in  the  first  example  following,  taking  the 
data  of  the  foregoing  example. 


184  COMPOUND    INTEREST 

ber  of  payments,  as,  for  instance,  the  supposed  pro- 
bability  of  the  life  of  the  person  enjoying  a  life  annu- 
ity, it  is  evident  that  the  only  difference  resulting 
would  be  in  the  number  of  the  years  which  denote  the 
power  of  the  ratio  of  the  series  of  the  payments. 
Suppose  it  should  take  place  m  years  after  the  lend- 
ing of  the  money,  or  beginning  of  the  compound  inte- 
rest upon  the  original  capital ;  we  would  then  have  : 

(1  +r)''-"_l 

a  =  C(l-t-r/-p 

r 

This  latter  is  usually  called  reversion. 

Example  1.  Supposing  the  capital  which  was  given  in 
the  preceding  section,  and  that  an  annual  pa3'ment  of 
^800  was  to  be  made,  beginning  with  the  first  year, 
and  letting  the  number  of  years  also  be  6,  we  shall  have 
the  amount  in  the  bands  of  the  receiver  of  the  money 
at  the  end  of  6  years  : 
By  the  expression 

(1,06)8—1 

a  =  7600  (1,06) «  — 800 

0,06 

a  =  $  10639,  9  —  6680,  266  =  $  6068,  633 

Example  2.  Suppose  the  same  capital  originally  given, 

and  the  same  payments,  to  begin  6  years  after  the  placing 

of  the  money  ;  what  will  be  the  amount  after  14  years  ? 

By  substituting  these  numbers  in  their  proper  place 

we  obtain  : 

(1,06)8  _  1 

a  =  7600  (1,06)14  _  SOO 

0,06 
from  which  is  obtained  : 

a  =  16966,88  —  1276,08  =  16681,8 
To  find  in  this  case  the  rate  per  cent,  or  the  number 
of  years,  having  given  the  other  parts,  will  again  require 
methods  of  calculation  which  lie  out  of  the  limits  of  this 
work,  as  may  be  judged  from  their  form,  and  by  refer- 
ence to  the  preceding  chapter  on  series. 


AWD    ANNUITIES.  185 

§  111.  The  determination  of  the  value  of  arrears 
of  payments  is  calculated  upon  the  same  principle 
as  the  payments  in  the  preceding  case;  because  it 
is  supposed  that  tlie  money  due  at  former  times,  and 
not  paid,  would  have  increased  in  the  same  manner; 
therefore  the  solution  of  these  cases  lies  in  the  second 
part  of  the  above,  and  the  result  is  obtained  by  a 
mere  change  of  denomination ;  thus  : 

The  amount  of  all  arrears  due  =  a 

The  yearly  payments  due  =  p 

I       The  rate  per  cent,  interest  =  r 

The  number  of  years'  arrears  due      =  n 

Gives  the  result  of 

(l+r)°-l 
o  ==  p 


xample.   An  annual  payment  of  $  1000  being  in  ar- 
rear  for  7  years,  what  is  the  amount  to  be  paid,  on  the 
principle  of  compound  interest,  at  the  rate  of  6  percent, 
annually  ? 
This  gives 

(1,06)7  —  1  0,503633 

a  =   1000 =  1000  . 

0,  06  0,  06 

or       a  =  g  8392,  22 

5  112.  When  a  certain  capital  is  to  be  distributed 
into  equal  pa^  ments  undei*  the  allowance  of  com- 
pound interest,  as  is  often  done,  the  expression  of 
^S  110  gives  the  principle  of  this  distribution  by  the 
simple  supposition  tliat  the  second  part  of  the  ex- 
pression, containing  the  amount  of  the  yeai'Iy  pay- 
ments, with  their  compound  interest,  must  be  equal 
to  tlie  first,  containing  the  capital  with  its  compound 
interest.     That  is  to  say,  we  have 

(l+r)«-  1 
C(l+r)«  =  p 

I* 


186      OOMPOFKTD    rJTTEREST  AITD  ANNriTIES. 

f-iliis,  considered  as  product  of  extremes  andmeanf? 
^^  a  geometric  proportion,  gives 

C  :p  =z  {\  -{-r)''  --  1  :  r  (l  +  rf 
So  we  may  determine  with  equal  ease  the  yearly 
payment  —  p,  which  will  extinguish  (or  be  equal 
to)  a  certain  present  amount  =  C,  at  the  rate  per 
cent  =  r,  in  the  number  of  years  =  n,  and  the 
present  capital  which  such  yearly  payments  will 
represent;  for  wfe  have  from  this  proportion  : 

r(l-frr 
p  =.  C ; 

(1  _f>r)"  — 1 

and  C  =^  p ; 

r  (1  -f-  ry- 
by  the  sin^ple  rule  of  three. 

In  substitutihg  here,  by  way  of  example,  the  numbers 
found  or  given  §  110,  the  abdv'e  expression  would  stan€l 
thus  : 

0.  06(1,06)« 

Payme^nt  g  800  =  6580,  2G6  — 

(1,06)8  —  1 

(1,06)8  —  1 

Capital  $  5580,  266  =  800 

0,06  (1,06)6 

The  determination  of  the  number  of  years  that  it 
will  take  to  extinguish  a  debt  by  given  yearly  and 
equal  payments,  is  another  question  that  is  beyond 
our  present  limits,  for  it  is  the  same  as  that  stated 
in  «5  108,  This  subject  is  therefore  dismissed,  and 
it  is  expected  that  any  student,  who  has  applied 
himself  to  this  exposition  of  the  principles  of  this 
kind  of  calculation,  witJi  tlie  necessary  understand- 
ing of  the  general  principles  of  arithmetic  taught 
in  this  book,  will  find  no  difficulty  in  solving  an} 
of  tlie  questions,  which  will  appear  at  the  end. 
upon  this  subject. 


AXLIGATIOS-*  iS7 


CHAPTER  IV. 

Of  Migation,  or  Mixtures  of  objects  of  different 
Value. 

§  113.  In  retail  mercantile  concerns  it  often  oc- 
curs, that  it  is  desirable  to  ascertain  tlie  proportional 
value  of  a  mixture  of  things  of  different  values  which 
are  given.  Reflection  upon  what  has  been  hereto- 
fore taught  would  point  out  the  principle  upon  which 
such  a  proportional  value  may  be  determined.  This 
value  of  the  mixture  being  naturally  a  certain  mean 
of  all  the  component  parts,  this  operation  of  arith- 
metic is  usually  called  alligation  medial. 

The  quantity  of  each  component  part  multiplied 
by  the  price  of  its  unit  (what  is  u>sually  called  its 
value)  evidently  gives  the  influence  of  this  part  up- 
on the  general  mixture.  It  might  therefore  be  con- 
sidered generally  as  acting  exactly  in  the  same  way 
as  the  product  of  cause  into  time.  The  sum  of  all 
these  products  evidently  constitutes  the  whole. 
Thus  we  might  say  in  any  number  of  things  mixed, 

CxT-f-cXi4-3X^-faXi  =  E 
the  sum  of  all  these  uniting  in  the  common  effect  =  E^ 
If  therefore  the  mean  eff*ect,  that  is,  the  mean  value 
of  each  individual  thing,^  or  unit,  in  the  miiXture, 
is  to  be  determined,  this  whole  effect,  that  is,  the 
sum  of  all  the  partial  effects,  is  to  be  divided  by  the 
juimber  of  things  mixed,  or  the  objects  acting  in  the 
general  result.  This  expressed  in  the  above  form 
will  give,  considering  C  for  the  cause  as  the  ob- 
ects  and  (the  time)  T  as  their  value,  the  following 
general  result : 

CX  T  +  cXt-^d  Xi-t-j  XX 
Mean  =         ■ 


Ib8  ALLIGATIOIf. 

Example,  Suppose  that  a  number  of  men  work  at  a 
certain  work  during  a  month,  as  follows,  namely  :  6  men 
work  15  days  each  ;  4  men  work  19  days  each  ;  12  men 
work  20  days  each  ;  and  10  men  work  26  days  each, 
during  that  time  ;  on  how  many  days'  work,  on  an  ave- 
rage, can  one  calculate  for  each  man  in  a  month  ? 

This  gives  : 

6X15-1-4  X  19-1-12X20+  10X26  13 

Mean  •= 20  H 

6+4  +  12+10  16 

In  this  manner  it  may  evidently  also  be  calculated, 
that  in  a  number  of  workmen  engaged  in  a  work  the  oc- 
casional absences  may  reduce  the  amount  of  work  which 
they  would  otherwise  perform  ;  to  the  mere  result  of  the 
product  of  the  denominator  of  the  above  fraction  into 
the  quotient  found,  or  the  above  workmen  taken  to- 
gether, would  in  a  month  have  executed  only  the  work 

W=  32  (20  +  if)  =  666  days  ; 
or  the  amount  of  the  numerator  of  the  fraction,  as  is  evi- 
dent ;  instead  of  which,  if  they  had  all  been  present  the 
whole  of  the  26  working  days  in  a  month,  they  would 
have  produced  the  work  =  W  =  26  X  32  =  832  days. 

§  114.  When  in  such  a  composition  it  is  desired 
to  obtain  a  certain  mean  value  of  the  objects  mixed, 
or  (as  in  the  preceding  example)  a  certain  amount 
of  work  by  means  of  objects  of  different  value,  (or, 
as  above,  men  differently  assiduous  to  their  work,)  it 
becomes  necessary  to  determine  the  quantity  of  eacli 
individual  ingredient,  (or,  as  above,  the  quantity  of 
each  men  of  a  certain  assiduity,)  to  obtain  the 
desired  aim,  that  is,  the  price  of  the  thing  aimed 
at,  (or  the  number  of  days'  work  desired.)  This 
operatioji  of  arithmetic  is  usually  called  alliga- 
tion alternate.  It  is  requisite  that  the  quantity 
of  objectsbelow  the  mean  value  must  compensate 
for  those  above  it;  their  products  must  therefore 
become  inverted.  In  thus  composing  a  mean  with- 
out limitation  of  the  quantity  to  be  made  up,  or  of 


AIXIGATION. 


18d 


%ny  of  the  parts  given,  it  is  evident  that  a  number 
of  solutions  will  be  possible  for  each  question,  but 
that  all  will  be  multiples  of  each  other.  The  prac- 
tical method  used  is  the  following. 

The  different  values  being  written  under  each 
other,  the  difference  between  one  value  above  the 
mean  and  this  mean  is  taken,  and  placed  oppo- 
site one  of  the  values  below  the  mean;  and  alter- 
nately, the  difference  between  this  lower  value  and 
the  mean  is  written  opposite  to  the  value  above  the 
mean;  thus  all  the  differences  being  taken,  the 
numbers  opposite  to  each  value  are  added,  and  give 
the  quantity  to  be  taken  of  each  of  these  respective 
values,  the  products  of  which  into  the  values  to 
which  they  are  opposite  will  give  a  sum  answering 
a  compound  as  desired.  And  every  equal  multiple 
of  all  the  parts  will  also  give  an  equal  multiple  of 
the  whole.  (The  parts  compared  are  linked,  to 
show  the  operation.) 

Example.  A  goldsmith  having  gold  15  carats  fine,  19 
carats,  21  carats,  and  24  carats,  wishes  to  make  a  mix- 
ture 20  carats  fine  ;  how  much  of  each  has  he  to  take  ? 


20 


which  gives 

13X5-f.  19X  5  +  21x6  4-24X  6  =  20  X  22  =  440 
or  the  whole  mixture  being  22,  be  it  ounces,  grains,  or 
That  it  may,  there  must  be  in  it  5  of  the  15  carats  gold  ; 
5  of  the  19  ;  6  of  the  21  ;  and  6  of  the  24  carats  gold  ; 
which  evidently  bears  the  proof  of  giving,  when  20,  the 
mean  price,  is  multiplied  by  22,  ihe  whole  quantity  mix- 
ed, the  same  result  as  is  obtained  by  the  sum  of  the  indi- 
vidual products. 

§  115.  If  either  the  whole  amount  of  the  mixture, 
or  any  one  of  the  parts  to  be  mixed,  is  limited  to  a 
certain  quantity,  it  becomes  necessary,  after  the 
above  operation,  to  take  the  ratio  between  the  part 


15— 

— 

4+1=5 

19— 

— 

1  -f-  4  =  5 

21  — 

1+5  =  6 

24— 

5+1=6 

*90  AtLlGATIOlr. 

given  and  its  corresponding  number  in  the  above 
result,  to  make  all  the  other  numbers  in  the  like 
manner  proportional  to  their  corresponding  ones  in 
the  above  result. 

Isf  Example.  If  in  the  above  the  whole  mixture  was 
required  to  be  36,  instead  of  22,  we  should  have  to  make 
the  proportions 

(the  15  carats,  or)  8,46 
("    19     "  ) 


22  :  36  = 


('*    21      "  )  9,818 

(" 


rirvc    €X   nil 


24     "  )  9,818 

2d  Example.  A  goldsmith  has  silver  6  ounces  fine,  10 
ounces  fine,  and  20  ounces  of  silver  9  ounces  fine  ;  how 
much  of  the  two  first  must  he  add  to  the  20  ouucea  of  9 
ounces  fine,  to  make  a  mixture  8  ounces  fine  ? 
1+2  =  3 
1  =   1 

1  c=     1 

This  will  give  the  ratio  of  the  silvers  ;  now  the  silver  at 
9  ounces  fine  being  determined  at  20  ounces,  the  propor- 
tion formed  from  the  ratio  of  the  number  found  for  that 
kind  of  silver,  to  the  number  limited  for  it,  is  that  which 
must  guide  all  the  others,  as  follows  : 

1  :  20  =  3  :  (silver  of  7  ounces  fine  =)  60 
1  :  20  =  1  :  (       "        10  "  =)  20 

§  116.  We  shall  now  close  these  elements  of 
arithmetic ;  for  to  go  into  more  complicated  practical 
applications  would  exceed  the  proper  limits  of  first 
elements,  and  may  be  much  better  treated  algebra- 
ically. The  regula  falsi,  or  rule  of  false  supposi- 
tion, both  simple  and  compound,  is  intentionally 
omitted,  the  first  because  an  attentive  scholar  of 
what  has  been  here  taught  will  not  need  it,  but  find 
in  what  he  has  learnt  the  better  means  to  solve  the 
question,  the  second  because  its  operations  belong 
more  properly  to  algebra,  so  far  as  they  actually 
lead  to  a  determined  result. 

§  1 1 7»  A  short  retrospective  view  of  what  has 
been  treated  in  these  elements  may  not  be  misplaced* 


KETBOSFECTIVE   VIEW.  t^f 

I  have  dwelt  at  some  length  upon  the  very  first 
elementary  ideas  of  arithmetic,  the  notation  or  signs 
of  the  arithmetic  operations,  and  the  principles  of 
the  systems  of  numeration,  because,  as  was  there 
said,  these  first  elementary  ideas,  if  well  under- 
stood, will  be  of  the  greatest  utility  in  rendering 
every  operation  in  arithmetic  easy  ;  it  is  therefore 
to  be  wished,  that  the  teacher  extend  them  still  more 
by  some  practice  upon  other  systems  of  numeration 
resides  the  decimal  system,  and  by  familiarising 
the  varied  combination  of  the  signs  of  arithmetic, 
the  full  value  of  these  combinations  being  ultimately 
assigned.  The  same  reasons  dictated  to  me  the 
detailed  de  scription  of  the  four  rules  of  arithmetic, 
which  it  is  certainly  proper  to  make  easy,  and  satis- 
factory to  the  mind  of  the  beginner,  if  he  is  ever  to 
know  how  to  apply  them  in  their  proper  place. 

In  treating  vulgar  fractions,  I  considered  it  obli- 
gatory upon  me  to  proceed  by  exact  mathematical 
demonstration,  and  to  deduce  them  from  their  actual 
origin  in  an  unexecuted  division ;  while  in  decimal 
fractions  the  whole  of  their  principles  will  at  once 
spring  from  the  consideration  of  division  conti- 
nued below  the  unit,  according  to  the  same  sys- 
tem as  above  it.  In  considering  all  conventional 
subdivisions  of  the  units  of  different  kinds  of  quan- 
tities as  denominate  fractions,  I  found  it  possible 
to  treat  it  with  some  system,  which  is  not  possible 
when  each  is  treated  separately.  If  I  have  deviated 
in  these  considerations  from  the  usual  method,  I 
hope  the  clearness  that  results  will  excuse  me.  It 
appeared  to  me  proper  to  bring  the  scliolars  to  tliis 
point  by  what  might  be  called  theoretical  steps. 

The  Second  Part  will  afford  the  scholar  the  satis- 
faction of  a  useful  and  amusing  application  of  the 
principles  learnt  before  I  considered  it  proper  to 
devote  a  separate  part  of  the  book  to  this,  in  order  to 
give  the  scholar  the  satisfaction  of  seeing  how  much 
he  could  do  with  the  few  elements  he  had  learned 


J 92  RETKOSPECTnrB   VIEW. 

before ;  and  it  is  to  be  hoped  that  every  teacher  will 
know  how  to  relieve  his  scholar  in  an  agreeable 
manner  by  this  Second  Part,  and  the  questions  which 
will  be  placed  hereafter,  or  others  of  his  own  making. 

In  the  Third  Part,  treating  of  ratios  and  propor- 
tions, 1  considered  myself  both  bound  by  true  prin- 
ciple, and  authorised  by  the  progress  of  the  scho- 
lar, to  treat  the  subject  as  the  beginning  of  the 
elements  of  the  actual  science  of  quantity ;  the  prin- 
ciples being  so  few  and  simple,  the  task  appeared 
to  me,  only  to  lay  them  well  open  to  the  scholar, 
and  to  show  him  all  their  bearings  and  conse- 
quences ;  a  defective  treatment  of  this  part  of  arith- 
metic, cannot  but  destroy,  instead  of  cultivating, 
the  reasoning  and  understanding  of  the  scholar. 
These  reasons  determined  me  to  a  more  detailed 
application  to  examples  fully  worked  out,  as  they 
both  help  to  explain  the  principles,  and  make  their 
application  pleasant  to  the  scholar. 

The  use  of  letters  to  denote  a  quantity  before  its 
determination  appeared  to  me  proper  to  be  intro- 
duced, and  gradually  to  habituate  the  scholar  to 
more  general  considerations  in  regard  to  quantity, 
not  servilely  attached  to  the  figures  of  our  system 
of  numeration. 

After  the  steps  made  in  the  Third  Part,  I  hope  to 
need  no  excuse  for  the  greater  degree  of  generalisa- 
tion which  has  been  introduced  in  the  Fourth,  except 
to  say  that  it  was  done  with  the  avowed  intention  of 
leading  the  scholar  imperceptibly  into  the  entrance 
of  algebra.  It  is  absolutely  useless  to  teach  these 
parts  by  rules ;  no  scholar  ever  remembers  them ; 
and  he,  whose  memory  is  mechanical  enough  for 
this,  seldom  knows  where  they  are  applicable. 
They  are  therefore  useless  to  him;  and  to  omit 
teaching  properly  the  principles  of  these  parts  is 
an  injustice  towards  the  student  of  f^rithmetic,  who 
wishes  to  prepare  himself  by  it  for  higher  studies. 


COLLECTION  of  QUESTIONS. 


I 


NtTMERATIOJT. 

Read  the  following  numbers : 

Ist.                     73,064;  6th.  94,070,790 

2a.            101,070,101;  7th.  4,399,080,502 

3d.                    500,007;  8th.  100,010,007 

4th.    90,  807,  060,  501  ;  9th.  7, 070,  409 

5tb.       1,897,510,234;  10th.  1,902,010,571 

ADDITION. 


Add  the  following-  numbers : 
Ist.  1, 006, 052  +  70,  401  -|-  8,  040,  107  -I-  9, 080,  071, 402  =^ 
2d.  17,040,109  -f-  50,201  -f-  701  -f  30  -f-  5.  000, 127  = 
3d.  70904  +  398125  +  8079123  -f-  98162753  = 
4th.  37  -f  90005  +  1009645  4-  309047  = 
.5th.  773  -I-  104462  -f-  34983  -f  81090406  = 

EXAMPLES    IN   MULTIPLICATION. 

1.  Seven  boys  have  each  twelve  marbles;  how  many  marbles 
have  they  altogether? 

2.  If  5  boys  buy  each  half  a  peck  of  apples,  and  each  half 
peck  holds  on  an  average  16  apples,  how  many  apples  have  they 
altogether  ? 

3.  A  company  of  soldiers  of  105  men  with  the  officers,  having  all 
muskets,  each  weighing  5  pounds,  and  2  pounds  of  ammunition, 
how  much  weight  have  they  to  carry  altogether  ? 

4.  A  ton,  ship's  weight,  is  2200  pounds;  how  many  pound? 
weight  will  be  in  a  vessel  carrying  450  tons? 

5.  Twenty  bales  of  cloih,  containing  each  27  pieces,  of  2C 
yards  the  piece,  how  many  yards  are  there  in  the  whole? 


DIVISION. 

1.  I  have  750  pieces  of  cloth,  and  can  put  no  more  than  13 
pieces  in  a  bale ;  how  many  bales  shall  I  have  to  make  ? 

2.  A  schoolmaster  has  62  boys,  and  having  a  lot  of  434  marbles 

17 


194  ctrESTioifs. 

Tvhich  he  wishes  to  distribute  equally  among  his  boys  as  arewarff^ 
how  many  will  each  of  them  get  ? 

3.  If  a  man  has  an  annual  income  of  ^3555,  how  much  can  he 
spend  per  day  ? 

4.  A  man  having  two  hundred  and  fifty  miles  to  travel,  and  tra- 
velling 24  miles  per  day,  how  long  will  he  be  in  performing  the 
journey  ? 


VULGAR  FRACTIONS. 

ADDITION. 

7  3        5         3         9         12 

1.  Add    -H 1 1 { h— = 

8  7        9        14       24        15 

9  2        7         8         3         9 

11        5        9        n         25        32 
1        7        2        17        19        15        16 

5        8        9       27        32        38        42 
9         15        13        14         8  10         5 

4.    "        -  + --  +  ---f- +  -  +  --{- ~=- 
13        19        21        27        23        34        18 

SUBTRACTION. 

Make  the  diflference  between  the  following  fractions,  added  an® 
subtracted  as  indicated  by  the  signs. 

343  7  936         11        2 

1. +  -  + +  -4- 

7       5       11        12         14       8        35        12       15 
42325  5         7         114 

7  9    14   15    8   12   16    18   5 

3427    3    6    6    9i 
3.   .4. -f 4. 4-_ 

8  5   9   15    14   21   25   26   3 

TO   FIND    THE    GREATEST    COMMON   MEASURE 

24598  74844 


2. 


44226  150579 

61047 


3. 


77373 


XO  riS3  THE  8VCCESSIVS  ArPROXIMATmO   BBACTI0K8* 
794973  5967 


1. 


1674219  13843 

38126  81097 


2.    :      4. 


2, 

» 

3. 

»» 

4. 

r 

5. 

»» 

$. 

V 

7. 

« 

«. 

)> 

516412  649321 


DECIMAL  FRACTIONS. 

REDCCTI0N   TO    DECIMAL    FRACTIOUS. 

1.  Reduce  13h.  7m.  into  decimals  of  the  day. 
56d.  7h.  into  decimals  of  the  year. 
10,  5  inches  into  decimals  of  the  foot. 
5oz.  7  dwt.  3§:r.  troy  into  decimals  of  the  pound. 
75  lb.  7  oz.  into  decimals  of  the  cwt.  avoirdupois. 
2  ft.  5,  7  in.  into  decimals  of  the  yard. 
27  h.  5  m.  3  s.  into  decimals  of  the  year. 
17  cubic  inches  into  decimals  of  the  cubic  foot. 

ADDITION. 

A  grocer  making  an  inventory,  finds  he  has  in  cash  $  17>  52;  in 
various  liquors  the  amount  of  $  215,  17 ;  in  soap,  candles,  and  such 
articles,  |  92,  54 ;  in  spices,  |  107, 32 ;  in  salt  fish  and  similar  pro- 
visions, $  49,  62 ;  and  in  various  small  articles,  besides  the  furni- 
ture of  his  store,  in  all  $  57, 84 ;  what  is  the  whole  amount  of  his 
stock  ? 

SUBTRACTIOW. 

t.  Subtract  as  follows :  7, 0107605  —  4,  901979865 

2.  '»  "  35,0964-34,9895602 

3.  "  "  670,4801—669,94013 

4.  "  "  0,04217  —  0,03948 

5.  "  "  0,  9080706  —  0, 8950326 

MULTIPLICATION. 

1 .  Bought  I7f  yards  of  cloth  at  $  2,  65  per  yard;  how  much  is 
the  amount  to  pay  ? 

2.  Multiply  10,09562X7,8059 
0,  00867X  9,  0472 
9,  80604X0,  0976 

301,0605X0,003908 
7503,09706X0,0009801 

DIVISION. 

6,0453  36,45097 

I.  Divide    =  :      2.  Divide 


3. 

n 

4. 

5» 

6. 

'> 

6. 

»> 

9^8106  0,00438 


196 


ftUBSTlONS. 


52, 0096  3, 09042 

2.  Divide =  ;       6.  Di?ide = 

6,  49502  95,  763 

0, 00652  655,  3708 

3.  " ==  ;      7.       '>        - 

3,4096  '         '  942,01)7 

0,  0043106  0, 04609 

4.  " =  ;       8. 


0,  09459  0, 000762 

MIXED    aUESTIONS   IN    DECIMAL   FRACTIONS. 

1  What  do  5  pieces  of  cloth  of  28i  yards  each,  come  to,  at 
|3,  37^  per  yard  ? 

2.  One  pound  sterling  is  equal  to  $4, 444 ;  (with  continued  de- 
cimals of  4;)  how  much  is  £975^,  expressed  iu  dollars  ? 

^ns.  14335,55122. 

3.  A  captain  of  a  vessel  has  on  board  706  packages,  each  mea- 
suring 1-8  of  a  ton  ;  89  others,  each  measuring  |  a  ton  ;  and  405 
others,  each  measuring  |  of  a  ton ;  how  many  tons  of  lading  has 
he  ?  ^ns,  264|  tons. 

3.  A  captain  has  on  board  170  bales,  each  paying  freight  |1,25  ; 
305  packages,  each  paying  87|  cents ;  230  tons  of  other  goods, 
each  ton  paying  $12,  62^;  and  6  passengers,  each  paying 
.$78,60;  how  much  does  bis  whole  freight  and  passage  money 
amount  to?  ji ns.  $2Q54,12h 

5.  A  raft  contains  305  pieces  of  timber ;  of  these  120  are  oak, 
36  feet  long  and  16  inches  square;  50  pieces  of  oak,  45  feet  6 
inches  long,  and  18  inches  by  14  inches  on  the  sides  ;  166  pieces  of 
pine  masts,  reckoned  at  2  feet  6  inches  square  and  60  feet  long. 
The  rest  pine  timber,  17  inches  square  by  50  feet  in  length.  The 
oak  timber  sells  at  45  cents  per  cubic  foot ;  the  masts  at  80  cents 
the  cubic  foot,  and  the  pine  timber  at  15  cents  the  cubic  foot. 
How  much  money  will  the  whole  raft  come  to  in  the  sale  ? 

6.  For  plastering  a  wall  the  mason  has  to  receive  21  cents  per 
square  yard  (or  the  square  of  3  feet  each  way,  and  containing  there- 
fore 9  square  feet ;)  the  wall  which  he  has  plastered  is  13i  feet  high, 
and  22  feet  long ;  how  much  has  he  to  receive  for  it  ?  J^ns.  |6,93, 
And  how  many  yards  does  the  wall  contain  ?  j^ns.  33  yards. 

7.  How  many  square  feet  front  of  brick  wall  can  be  built  with 
3600  bricks,  the  thickness  of  the  wall  being  the  length  of  two 
bricks,  and  the  end  of  the  bricks  being  four  inches  by  two? 

Jins.  1000  feet  square. 

8.  A  merchant  makes  16i  per  cent,  upon  merchandise  that  costs 
him  |7, 65  ;  how  much  will  his  profit  amount  to  ? 

=  7650  XO,  165  =  ^ns.  $1262, 25,  (according  to  the  princi- 
ples of  decimal  fractions.) 

9.  The  tare  allowed  upon  a  certain  merchandise  is  2i  per 
cent. ;  how  much  will  it  amount  to  upon  7355  weight  ? 

(Expressed  as  ^bove)  =  283, 875. 


(^VESTIONS.  19T 


DENOMINATE  FRACTIONS. 

ADDITIOir. 

1.  Add  J67  6s.  7d.  -f  £3  4s.  lOd.  -{-  Ss.  4d.  -|-  £9  14s.  lid, 
-4-  £23  17s.  5d. 

2.  Add  3  lb.  4  oz.  17  dwt.  5  gr.  -|-  15  dwt.  17  gr.  -f  17  lb. 
3  dwt.  4  gr.  4-  17  oz.  15  gr.  -4-  31b-  12  dwt.  6  gr. 

3.  Add  6  yds.  2  ft.  3, 4  in.  +  17  yds.  5  iu.  +  22  yds.  1  ft.  11  in. 
-J-  62  yds.  1  ft.  9  in.  -{-  34  yds.  10  in.  -\-  69  yds.  2  ft.  9  in. 

4.  Add  7  miles  3  farlongs  17  yds.  -j-  21  m.  1  fur.  30  yds.  + 
S4m.  3  yds. 

5.  Add  24  bush.  3  pecks  +  19  bush.  5  pecks  +  18  bash.  2 
pecks  -{•  42  bush.  1  peck. 

SUBTRACTIOir. 

1.  A  grocer  had  according  to  his  last  inventory  317  lb.  10  oz.  of 
sugar;  561  lb.  4  oz.  of  coffee;  451  lb.  6  oz.  tea;  15  lb.  3  oz.  pep- 
per; 3  oz.  6  dwt.  mace;  152  lb.  rice;  17  gallons  rum.  He  has 
sold  since,  283  lb.  6  oz.  sugar  ;  341  lb.  7  oz.  coffee  ;  349  lb.  5  oz. 
tea;  11  lb.  8  oz.  pepper  ;  2  oz.  6  dwt.  mace  ;  5  gallons  and  3  gills 
of  rum  ;   121  lb.  7  oz.  rice ;   how  much  has  he  left  of  each  kind  .' 

2.  A  man  has  to  travel  75  miles;  he  walks  thejfirst  day  20  miles 
3  furlongs  ;  the  second  18  miles  5  fur.  20  yds. ;  the  third  23  miles 
7  fur.  50  yds. ;  how  much  of  his  journey  remains  eyery  evening 
to  be  performed  .'' 

3.  William  the  Conqueror  acquired  the  throne  of  England  the 
26th  December,  1066,  and  died  8th  September,  1087.  '  His  son 
William  the  Second,  who  immediately  succeeded,  died  the  2d 
August,  1100.  Henry  the  first  succeeded,  and  died  the  1 0th  De- 
cember, 1 1 35      How  long  did  each  of  them  reign  .•* 

4.  Three  men,  starting  at  the  same  time  from  one  place,  arrived 
at  another  determined  place,  the  first  after  10  h.  16  m. ;  the 
second  after  12  h.  42  m. ;  the  third  after  15  h.  3  m.  How  much 
did  each  of  them  arrive  after  the  other  ? 

MULTIPLICATION. 

1.  Bought  27  lb.  5  oz.  16  dwt.  of  drugs  at  the  rate  of  $9,  75  the 
pound;  how  much  will  be  the  amount  ? 

2.  Bought  three  bales  of  cotton,  the  first  weighing  1016  lb.,  the 
second  998  lb.,  the  third  1093  lb.,  at  17^  cents  the  pound  ;  what 
is  the  amount  to  pay? 

3.  A  room  is  22  feet  5  inches  long  and  18  feet  9  inches  broad  ; 
how  many  yards  of  carpet  will  it  need  ? 

4.  A  wall  is  8  feet  7  inches  high,  and  65  feet  9  inches  in  circum- 
ference ;  how  many  feet  of  plastering  will  be  in  it  .'* 

5.  Required  the  solid  contents  of  a  wall  74  feet  6  inches  long, 
2  feet  9  inches  broad,  and  24  feet  4  iachee  high .'' 

17* 


^98  ^ITEITIONS. 

6.  Required  the  solid  contents  of  a  box  5  feet  2, 5  inches  lonff. 
3  feet  5  inches  broad,  and  2  feet  5,8  inches  deep  ? 

6.  How  many  cubic  feet  of  earth  will  fill  a  dock  205  feet  longf 
75  feet  broad,  and  8  feet  7  inches  deep  ? 

DIVISION. 

1.  If  87  lb  6  oz.  of  coffee  cost  $18,  38,  what  is  the  price  of  one 
pound  ? 

2.  What  is  the  price  per  pound  of  spices,  when  34  lb.  7  oz.  cost 
^25,  82  ? 

3.  What  is  the  length  of  a  piece  of  timber  15  inches  square,  the 
cubic  contents  of  which  is  69  feet  6  inches  ? 

4.  What  must  be  the  depth  of  a  square  vessel,  1  foot  3  inches 
one  way,  and  2  feet  2,5  inches  the  other  way,  that  shall  hold  4  feet 
2,5  inches  cubic  measure  ? 

5.  What  must  be  one  side  of  an  area  containing  2015  square 
feet,  when  the  other  side  is  50  feet  7  inches  ? 

6.  If  a  horse  runs  8  times  around  a  circus  in  1  h.  45  m.  20  s^,. 
how  much  time  will  it  need  for  each  turn  ? 

7.  A  lumber  merchant  bought  6527  cubic  feet  of  timber,  in  321 
pieces  ;  how  much-did  each  piece  averaa;c  in  cubic  feet  ? 

8.  A  brick  wall,  two  bricks'  length  in  thickness,  is  69  feet  long 
and  26  feet  high ;  how  many  bricks  does  it  contain,  each  brick 
being  8  inches  long,  4  inches  broad,  and  2  in^thes  thick,  when  laid  ? 

PRACTICAL  QUESTIONS  FOR  THE  SECOND  PART. 

1.  A  purchase  of  goods  that  cost  ;^765,25  was  sold  for  ^973,52 ; 
what  was  the  profit  ? 

2.  A  man  has  ;^8264,91  debts,  and  his  property  amounts  to 
j^743l,80;  how  does  he  stand? 

3-  Three  men  buy  land,  the  one  5,212  acres,  at  ^2,25  per  acre, 
the  other  bought  281  acres  for  ^600,  and  the  third  bought  as  much 
land  as  they  both,  for  ;^892 ;  what  had  the  first  to  pay,  how  much 
land  did  the  second  buy,  how  much  land  had  the  third,  and  at 
what  price  did  it  stand  him  ? 

4.  A  brick,  when  laid  in  the  wall,  has  7,8  inches  length,  3,{> 
breadth,  and  1,8  inches  thickness  ;  how  many  bricks  will  it  take  to 
build  a  wall  two  lengths  of  bricks  thick,  25  feet  long,  and  36  feet 
high  ? 

5.  At  6  per  cent,  interest,  what  must  be  the  capital  that  will 
produce  an  income  of  ;g;500? 

6.  A  man  having  $600  a  year,  how  much  may  he  spend  a  day 
to  save  ^200  in  the  year  ? 

7.  What  is  the  interest  at  7  per  cent,  of  ;g[12,450? 

8.  Upon  20  hogsheads  of  Sugar,  of  850  lbs.  each,  what  is  the 
tare,  at  3  lb.  for  every  hundred  weight  ? 

9.  Three  persons  purchase  together  ^500  of  stock,  at  5  per  cent, 
premium,  which  brings  in  8  per  cent,  interest;  how  much  must 
each  pay,  and  how  much  yearly  interest  will  each  have  for  his 
share  ? 


(QUESTIONS.  19§ 

iOi  A  house  is  to  be  plastered,  at  21  cents  per  square  yard. 
Now  there  has  been  plastered  an  entry  35  feet  loug,  and  1 1  feet 
6  inches  high  on  both  sides,  the  2  ends  being  given  in,  as  compen- 
sation for  the  vacancies  on  the  sides.  Two  roon;8  of  the  same 
height,  in  each  of  which,  two  sides  of  20  feet  long  are  reckoned 
full,  and  one  end  of  18  feet  also  reckond  full,  to  compensate  for 
the  vacancies,  the  fourth  side  is  given  iu.  Two  upper  rooms  of 
20  feet  long,  14  feet  broad,  and  9  feet  high,  are  reckoned  in  the 
same  manner  as  those  below,  and  one  room  has  14  feet  by  16  feet 
6  inches,  which  is  considered  as  plastered  all  round.  How  much 
will  the  expense  of  the  whole  plastering  be  ? 

11.  Suppose  the  above  entry  and  rooms  were  to  be  wainscoted 
with  simple  boards,  at  the  rate  of  1 1,23  for  every  hundred  square 
feet,  what  would  be  the  expense  ? 

12.  A  quantity  of  goods  is  bought  for  ^3,521,  and  sold  at  15 
per  cent,  loss,  for  what  was  it  sold  ? 

13.  A  dock  to  be  filled  in,  has  250  feet  length,  95  feet  breadth, 
and  the  perpendicular  depth  being  8  feet  on  an  average,  how 
many  cart  loads  of  earth  are  needed  to  fill  it,  at  the  rate  of  7  cu- 
bic feet  for  a  cart  load  ;  and  how  much  will  it  cost  at  6  cents  per 
load  ? 

14.  How  much  will  the  glazing  of  a  house  cost,  that  has  28 
windows,  each  of  24  panes  of  glass,  at  the  rate  of  13i  cents  for 
each  pane  ? 

15.  How  many  bricks  are  there  in  a  wall  two  lengths  of  a  brick 
thick,  20  feet  long,  and  38feet  high,  the  bricks  being  of  the  dimen- 
sions stated  in  the  fourth  question  ? 

16.  An  old  tower  40  feet  square  on  the  outside,  has  at  first,  a 
wall  10  feet  thick  for  20  leet  of  elevation,  then  for  36  feet  the  wall 
is  8  feet  thick,  then  for  16  feet  it  is  5  feet  thick,  the  outer  sides  be- 
ing perpendicular;  how  many  cubic  yards  of  stone  are  there  in 
these  walls,  (neglecting  doors  and  window  opeoiwjjs)  how  much 
will  the  stones  cost,  at  22  cents  the  cubic  yard,  and  how  much 
will  the  building  of  the  wall  cost,  at  the  rate  of  29  cents  for  every 
cubic  fathom  ?  What  will  be  the  weight  of  stones  in  it,  the  cubic 
foot  being  reckoned  at  178  lbs.? 

17.  A  carpenter  has  6i  cents  pef  cubic  foot  for  hewing  timber : 
now  he  hewed  25  pieces  of  15  inches  square,  (on  each  side)  and 
36  feet  long;  16  pieces  of  one  foot  each  way,  Hud  42  feet  long; 
28  pieces  18  inches  by  20,  and  26  feet  long  ;  12  pieces  of  10  inches 
each  side,  and  32  feet  long  ;  and  15  pieces  of  8  inches?  by  12  each 
side,  and  18  feet  long.     How  much  money  has  he  earned  ? 

18.  Two  rooms  are  to  be  painted  all  round,  the  height  of  which 
is  12  feet  4  inches,  the  length  of  one,  32  feet,  and  its  breadth  24 
feet ;  the  other,  18  feet  6  inches  long,  aud  16  <eet  5  inches  broad, 
how  much  will  be  the  cost,  ai  7  cents  per  square  yard  ? 

19.  What  will  be  the  expense  of  paving  a  street  563  feet  long, 
and  30  feet  wide,  at  the  rate  of  65  cents  per  square  yard  ? 

20.  What  will  be  the  weight  of  lead  that  is  upon  a  roof  25  feet 


200  (QUESTIONS. 

long,  and  28  feet  6  inches  slant  on  each  side,  at  the  rate  of  8i  lbs, 
the  square  foot  ? 

21 .  What  will  be  the  amount  of  slating  a  roof  of  38  feet  6  inches 
long,  31  leet  4  inches  slant  on  each  side,  at  the  rate  of  |4,25  per 
square,  of  10  feet  side  ? 

22.  How  many  days  will  three  carpenters  take  to  shingle  a  roof 
S8  feet  long,  28  feet  slant  on  one  side,  and  32  on  the  other,  at  the 
rate  of  two  and  a  half  square,  of  10  feet  side,  per  day  for  each 
man,  and  how  much  will  it  cost  at  |1,20  per  square  ? 

23.  What  will  be  the  amount  of  4572  square  feet  of  boards,  at 
the  rate  of  ^10,50  per  thousand  feet. 

24.  A  vessel  imports  goods  to  the  amount  of  ^9650,  which  pay 
duties  at  21  per  cent,  on  their  value ;  of  $12,600,  paying  30  per 
cent. ;  and  of  ^21580  pay  IS  per  cent,  duty  ;  besides  30  casks  of 
•wine,  averaging  58  gallons,  each  of  which  pays  20  cents  per  gal- 
lon ;  what  will  the  duties  on  the  whole  cargo  amount  to  ? 

25.  How  many  miles  did  that  vessel  travel  in  a  year,  which 
made  three  times  the  voyage  to  Europe  and  back  again,  every 
time  averaging  26  days,  sailmg  in  a  mean,  at  the  rate  of  6i  miles 
an  hour  ? 

26.  If  a  voyage  to  Batavia  takes  ninety  days,  the  vessel  sailing 
on  an  average  5|  miles  an  hour,  how  many  miles  does  the  vessel 
sail  in  the  whole  voyage  ? 

27.  If  a  baker  works  out  9  barrels  of  flour  every  working  day  in 
the  year,  at  196  lbs.  each  barrel,  how  many  pounds  of  flour  does 
he  use,  and  if  he  make  one  third  more  weight  of  bread  out  of  it, 
how  many  pounds  of  bread  does  he  make,  and  if  he  sells  the  bread 
at  4  cents  the  pound,  how  much  does  he  make  in  a  year,  when 
the  flour  cost  ^5  per  barrel? 

28.  If  18  dozen  bottles  of  wine  coat  |62,  what  is  the  price  of 
each  bottle  ? 

29.  The  nearest  approximation  between  the  earth  and  Venus, 
is  in  a  meau  32,560,000  miles,  the  velocity  oJ  a  cannon  ball  being 
about  2000  feet  in  a  second,  how  long  would  tde  cannon  ball  have 
to  run,  to  go  from  planet  to  planet,  if  they  remained  stationary  in 
such  a  position  ? 

30.  A  years  rent  of  a  house  being  $96,  the  occupant  has  laid 
out  in  repairs  ^24,56,  and  paid  the  taxes  amounting  to  $7,45,  what 
has  he  yet  to  pay  ? 

31.  A  man  having  $660  a  year,  economises  |150  annually;  his 
income  being  raised  to  $1500  a  year,  how  much  can  he  spend 
daily  to  economise  double  as  much  as  before  ? 

32.  If  a  man  earns  65  cents  per  working  day.  at  what  price  can 
he  board,  so  as  to  save  $89  for  his  clothing  and  o  her  expenses  per 
year  ? 

33.  A  bill  of  Exchange  on  London  for  jG372  129.  sterling,  is 
bought  at  8  per  cent,  premium,  what  is  to  be  paid  for  it  in  dol- 
lars, at  $4,44  the  £, 

34.  What  will  the  commission  at  2|  per  cent,  amount  to  oti 
goods  of  the  amount  of  $7652. 


(^UBSTIOJrS.  201 


QUESTIONS  IN  THE  RULE  OF  THREE. 

1.  A  merchant  bought  795  yards  of  cloth  for  |107  J,50,  he  has 
still  |427,50  which  he  wishes  to  lay  out  in  the  same  cloth,  at  the 
former  price  ;  how  many  yards  may  he  yet  purchase  ? 

2.  If  the  matting  for  the  floor  of  a  room  24  feet  by  18,  cost 
|95,60,  what  will  the  same  matting  come  to,  for  a  room  22  feet  in 
lengfth,  by  38  in  breadth  ? 

3.  How  many  yards  of  paper  22  inches  broad,  will  cover  a  wall 
of  26  yards  circuit,  and  9  feet  high,  if  20  yards  circuit  of  the  same 
height  can  be  covered  by  72  yards  of  30  inch  wide  paper  ? 

4.  It  takes  to  clothe  a  regiment  of  750  men,  5920  yards  of  yard 
wide  cloth,  how  many  yards  of  cloth  of  1  5-8  yards  wide,  will  it 
require  to  clothe  the  same  ? 

5.  The  forage  required  by  a  body  of  cavalry,  for  a  month  of  31 
days,  is  2821  cwt.  of  hay,  how  much  will  be  needed  for  the  same 
body  for  87  days  ? 

6.  If  172  boards,  17  feet  6  inches  long,  and  14  inches  broad,  are 
needed  to  floor  a  place,  how  many  would  it  take  12  feet  8  inches 
long,  and  10  inches  broad  ? 

7.  How  manypounds  of  tea  can  a  man  buy  for  |672,  if  he  buys 
751  lbs.  for  |327,50  ? 

8.  If  21  men  could  perform  a  work  in  17  day?,  and  16  men  be 
added  to  them  after  the  second  day,  how  much  time  will  be  saved 
by  it? 

9.  The  common  step  of  a  horse  being  about  4  feet,  and  that  of  a 
man  2|  feet,  the  man  making  8  steps  to  the  horses  5,  how  much 
space  will  the  man  gain  over  the  horse,  in  walking  a  distance  of 
18  miles? 

10.  The  annual  wages  of  a  man  being  $100,  to  be  paid  in  lani 
at  $6  per  acre,  how  many  acres  will  he  receive  after  3  years  and 
7  months  ?  • 

11.  Two  men,  A  and  B  bought  together  200  acres  of  land,  each 
paying  |200  ;  they  divide,  and  A  making  choice  of  the  better 
land,  they  agree  to  value  his  land  at  $2,25  the  acre,  and  that  of 
B  at  $1,75  ;  how  many  acres  will  each  of  them  get  ? 

A  gets  87,5. 
B  gets  112,5. 

12.  If  the  interest  of  money  is  7  per  cent,  what  will  be  the  dis- 
count? ^  Ans.  6,54205608. 

13.  How  much  must  a  man  pay  down  to  receive  in  6|  years 
$658,  the  interest  being  7  per  cent.,  calculating  upon  simple  in- 
interest  ?  Ans.  $452,233. 

14.  On  the  importation  of  certain  goods,  a  merchant  gains  20 
per  cent,  when  the  duty  is  16i  per  cent.,  what  per  cent,  will  he 
gain  upon  the  same,  when  the  duty  is  raised  to  18  per  cent.? 

15.  Two  travellers,  A  and  B,  leave  two  places  100  miles  dis- 
tant from  each  other,  at  the  same  time ;  A  travels  6|  miles  per 


SOS  f^UESTIOWS. 

bour,  and  B  7|  miles  per  hour,  what  part  of  the  distance  will  each 
of  them  make  ?  ^„    ^  A  ==  44,9205. 

•^^i  B  =  55;0295. 
And  what  time  will  they  travel  before  they  meet  ? 

Ans.  7  h.  6  min.  nearly. 

16.  How  many  yards  of  cloth  were  there  in  a  piece  which  cost 
$66,60,  the  price  of  the  yard  being  to  the  number  of  yards,  as  5 
to  7?  ^ns.  9,5561. 

17.  The  sum  of  two  numbers  multiplied  by  the  greater  is  120, 
the  same  multiplied  by  the  less  is  105,  what  are  the  two  numbers  r^ 

Ans.  8  and  7. 

18.  The  slow,  or  parade  step  of  the  military  being  70  steps  per 
minute,  and  the  step  28  inches,  how  far  would  troops  travel,  by 
marching  8  hours  in  a  day  ? 

19.  The  hour  and  minute  hand  of  a  clock  are  together  at  12 
o'clock,  when  are  they  together  after  each  hour  afterwards? 

20.  Of  two  travellers  upon  the  same  road,  A  travels  5  miles  an 
hour,  B  3  miles  an  hour;  when  B  passes  a  certain  place  on  the 
way,  A  is  still  13  miles  behind  him ;  at  what  distance  will  he  over- 
take B  ?  Ans.  at  32i  miles. 

21.  Two  men  bought  a  lottery  ticket  in  partnership,  A  gave  |9 
towards  it,  B  gave  |7 ;  the  ticket  draws  a  prize  of  $2000,  how 
much  will  each  of  them  get. ^  ^      <  A  =  1125 

Ans.^^^    875 

22.  The  father  of  a  child  is  52  years  older  than  the  child,  his 
mother  36  years  older,  and  the  age  of  the  father  is  to  that  of  the 
mother  as  4  to  3,  what  is  the  age  of  the  child  ?  Ans.  12  ys. 

23.  The  age  of  a  man  and  his  wife  are,  together,  equal  to  49; 
if  6  be  added  to  the  age  of  the  man,  and  11  subtracted  from  that 
of  the  woman,  the  numbers  will  be  in  the  ratio  of  9  : 2 ;  what  are 
the  ages  of  each .''  a      S  ^^^  man's  age  =;=  30. 

*^"*'  X  The  wife's  age  =  19. 

24.  The  number  of  cattle  on  my  farm,  is  to  that  of  the  cattle  of  my 
neighbour,  as  2  to  3,  and  if  each  of  us  had  four  head  of  cattle  more* 
the  number  would  be  as  5  to  7  j  how  many  head  of  cattle  has  each  ? 

Ans.  16  and  24. 

25.  The  age  of  a  father  is  to  that  of  his  child,  a=«  9  to  2,  and  the 
father's  age  is  12  years  more  than  3  times  the  age  of  the  child  j 
what  are  their  different  ages  ?     >,  .  S  The  child's  age,  8  yrs. 

'^'^'  I  The  father's  age,  36  yrs. 

TO  EXTRACT  THE  SaUARE  KOOT. 

1,    To  extract  the  Square  Root  of    1296 


2. 

(( 

li 

(( 

« 

t( 

(( 

7921 

3. 

o 

{( 

u 

l( 

« 

(( 

9899 

4. 

tt 

(( 

<l 

(i 

« 

(i 

25,1001 

6. 

(( 

« 

l( 

(( 

(« 

(i 

6905,61 

d. 

(i 

<» 

ct 

(( 

u 

(« 

476991, 

7. 

i( 

it 

<t 

(( 

t« 

(( 

3, 

9. 

»( 

(( 

« 

(( 

u 

(( 

7, 

0. 

(( 

<( 

(( 

(( 

(( 

<t 

18,49 

<tT7BSVI017B. 

TO  EXTRACT  THE  CUBE  ROOT. 


1. 

Of    9261 

2. 

"     1906,624 

3. 

»<     20570824 

1                       4. 

**     4052,24 

5. 

"     43243551 

6. 

«'      103161,700 

^                7. 

«     1520,  ii75 

H 

»     216,000 

m 

«     5832,761 

^^                 10. 

«'     64,372 

CbUADRATIC    EaUATIONB. 

Given  a;  ^  —  8 

flc  —  7  =  13  to  find  a. 

jjfw.  10. 

"    3.:b  »  —2 

x  =  40  :       to  find  x. 

Ansy  4 

J*.    t^^~i 

x  4-  V  =  ?  to  find  a:. 

.5»w.  3. 

1. 

2. 
3. 

4.  To  divide  ten  into  two  parts,  so  that  their  product  shall  be 
equal  to  12  times  their  diflference.  Ans.  4  and  6. 

5.  To  divide  13  into  three  parts,  so  that  the  diflferences  between 
the  squares  shall  be  equal,  and  the  sum  of  the  squares  =  75. 

Am.  1;5;  7. 
Y  ARITHMETIC  PROGRESSION. 

1.  A  sets  out  from  one  place,  and  B  from  another,  360  miles  dis- 
tant from  A  :  they  travel  towards  each  other,  so  that  A  performs 
the  first  day  40  miles,  the  second  3U,  the  third  36,  and  so  on,  de- 
creasing his  rate  2  miles  daily.  B  bei^ins  the  first  day,  and  travels 
50  miles,  the  second  22,  increasing  his  rate  two  miles  every  day  j 
in  how  many  days  will  they  meet  i  Ans.  6. 

2.  Find  the  sum  of  the  series  of  natural  nunabers  up  to  100. 

Aris.  =  6050. 

3.  Find  the  sum  of  the  series  of  even  numbers  up  to  100. 

Am.  =2550. 

4.  Find  the  sum  of  the  series  of  odd  numbers  up  to  200. 

Am.  =  10,000. 

5.  Two  travellers  setting  oat  together  from  the  same  place,  A 
travels  the  first  day  8  miles,  and  increa?'es  hiy  rate  4  miles  every 
day.  B  goes  25  miles  per  day  froa'.  the  be^inain^^ ;  how  many  days 
will  they  be  in  meeting  ^ 'Am?  Anf.  15,5  days. 

6.  It  is  required  how  fiir  125  stones  roust  be'  placed  at  equal 
distance  from  each  other,  that  the  sum  of  iheir  distances  from  a 
point  20 yards  before  the  first,  maybe  >-xac{iy  equal  to  5 miles? 

7.  How  many  eg*s  will  be  neeueJ  to  lay  4  feet  apart,  to  occa- 
sion a  roan  who  has  to  pick  thera  uj*  oiie  by  one,  and  bring  them 
to  a  ba«ket  3  yards  behind  the  firsf,  to  hiive  to  walk  2,75  miles  ? 

8.  To  find  7  arithmetic  meaur  between  6  and  46. 

9.  How  many  strokes  does  a  clock  strike  in  one  whole  day,  by 
our  common  division  of  time,  striking  from  1  to  12. 


204  IIUESTIOKS. 

10.  A  man  having  to  pick  up  102  eggs  laid  in  a  row  on  the 
grouod  at  one  yard  from  each  other,  and  carry  them  in  a  basket 
at  two  yards  from  the  first,  while  another  has  to  walk  a  distance 
of  three  miles  from  the  same  place  and  back  asjain,  which  of  the 
two  has  the  advantage  in  the  distance  to  be  walked  through  ?  and 
how  much  ? 

Ans.  The  one  who  walks,  has  only  56  yards  more  to  walk. 

11.  If  one  cent  is  placed  on  the  first  square  of  the  chess  board, 
and  one  more  on  each  subsequent  square,  how  many  dollars  will 
be  upon  the  whole  board  ?  Ans  $20,80. 

GEOMETRIC  PROGRESSION. 

1.  What  is  the  sum  of  one  hundred  terms  of  the  powers  of  2 . 

2.  What  is  the  sum  of  30  terms  of  the  powers  of  3  ? 

3.  The  first  term  of  a  geometric  series  being  20,  and  the  ratio 
I,  what  will  the  25th  term  be  ? 

ALLIGATION. 

1  A  man  in  a  month  of  twenty-six  working  days  works  6  days 
at  the  rate  of  |1,15  a  day,  5  days  at  75  cent?  per  day,  3  days  at  $2 
per  day,  ten  days  at  $1,50  a  day,  and  is  idle  the  2  remaining  days  ; 
at  what  rate  per  day  does  he  earn,  counting  the  whole  of  the  30 
days  in  a  month  ? 

2.  What  is  the  fineness  of  a  mixture  of  2  oz.  of  gold  23  carats 
fine,  7  oz.  22  carats  fine,  9  oz.  17  carats  fine,  and  3  oz.  20  carats 
fine? 

4.  A  merchant  sold  a  quantity  of  cloth,  namely,  150  yards  at 
|3,75,  which  cost  him  $3  per  yard,  720  yards  at  $5  the  yard,  which 
cost  him  $3,75  per  yard,  305  yards  at  $7,50  per  yard,  which  cost 
him  $6,35  per  yard,  and  100  yards  at  $2,50,  which  cost  him  $2 
per  yard ;  how  much  did  he  make  per  yard  on  an  average  ? 

5.  A  mixture  is  to  be  made  of  silver,  some  of  which  has  cost 
$1,10  per  oz.,  some  97  cents  per  oz.,  and  the  rest  88  cents  per  oz. ; 
the  mixture  is  to  weigh  3  lbs. ;  how  much  is  to  be  put  in  of  each, 
to  make  the  intrinsic  value  of  the  silver  just  a  dollar  an  ounce  ? 

COMPOUND  INTEREST  AND  ANNUITY. 

1.  A  man  having  an  income  of  $5000  a  year,  rHves  one  quarter 
of  his  income  a  year,  which  he  puts  to  iiiiere<t :  what  will  be  the 
amount  of  his  savings  in  12  years,  at  compoimci  interest,  at  5  pei 
cent,  per  annum  ? 

2.  A  trader  living  at  the  yearly  expense  of  $500,  and  trading 
with  the  rest  of  his  stock,  augments  it  one  thi-i  every  year;  at 
the  end  of  the  third  year  his  stock  is  doubi^-d  ;  voHt  was  his  origi- 
nal stock  ? 

3.  If  a  man  spends  every  year  his  whole  >in.iual  income,  and 
one  quarter  of  that  sum  in  addition,  which  ho  t  ike^^  from  his  ca- 
pital bearing  interest  at  5  per  cent. ;  how  many  y^'ars  will  he  be, 
in  spending  the  whole  capital  itself,  from  the  first  beginning  ? 


Q,€ESTlONS.  905 

PROMISCUOUS  QUESTIONS. 

1.  The  sides  of  two  square  pieces  of  ground,  are  as  3  to  5,  and 
the  sum  of  their  superficial  content  is  30600  square  feet;  what  is 
(he  \ea^\.h  of  the  sides  of  each? 

^ns.  90  feet  and  150  feet. 

2.  Three  young  men  entering  into  partnership,  agree  to  make 
a  common  stock,  to  which  each  shall  contribute  in  the  ratio  of  the 
sum  of  the  ages  of  the  two  other  partners.  A  is  24  years  old,  B 
27  years,  and  C  31  years  ;  what  will  be  the  share  of  each  ? 

3.  A  parcel  of  tobacco  is  sold, .some  at  12  cents  per  pound,  the 
rest  at  15  cents  per  pound  ;  the  proportion  of  the  first  to  the  latter, 
was  as  4  to  |,  and  the  amount  of  the  sale  $380  ;  how  many  pounds 
were  there  of  each  kind  ?  -       V  Of  the  first,  1500  lbs. 

'^^-  \  Of  the  2d,     i:333j  lbs. 

4.  A  grocer  bought  cofiee,  3  bags  of  80  lbs.  at  21  cents  per  lb., 
6  bags  of  53  lbs.  at  i;4  cents  per  lb.,  and  9  bags  of  90  lbs.  each,  at 
18  cents  per  It).,  and  sold  the  whole  together  at  22  cents  the  pound ; 
what  did  he  make  by  it  .=•  j^ns.  $47,84. 

5.  What  fraction  is  that,  to  the  numerator  of  which,  if  1  be 
added,  it  becomes  ^,  and  if  1  be  added  to  the  denominator,  it  be- 
comes i  ?  Ans.  y*3.. 

6.  The  quick  step  of  troops  in  marching,  is  2  steps  of  28  inches 
each  in  a  second  ;  how  far  wid  such  troops  travel  in  a  day  of  eight 
hours? 

7.  The  captain  of  a  vessel,  of  which  he  owned  ^,  sold  out  the 
half  of  his  share  ;  he  had  before  the  sale  $350  annual  profit  from 
it,  besides  his  wages  ;  how  much  remains  to  him  annually  after  the 
sale  of  this  part  of  his  share  ? 

8.  A  draper  sold  from  a  piece  of  cloth,  i  at  $5  the  yard,  one 
fifth  at  $4  per  yard,  and  one  sixth  at  $4,50  per  yard ;  by  this  he 
obtained  $168 ;  how  many  yards  were  there  in  the  piece  ? 

^4ns.  60. 

9.  On  the  first  of  January  1793,  a  royalist  in  Europe  agreed 
with  a  democrat,  to  pay  him  3  cents  per  day,  until  the  restoration 
of  the  Bourbons,  on  condition  of  his  paying  him  one  louis  d'or  on 
$4,44,  every  day  after  that  restoration.  Taking  the  first  of  Au- 
gust, 1814,  for  the  day  of  the  return  of  the  Bourbons,  how  would 
their  account  stand  on  the  first  of  January,  1827,  omitting  all  in- 
terest ;  and  how,  on  calculating  compound  interest  for  every  day 
from  the  epoch  of  these  payments,  to  the  first  of  January,  1827,  at 
5  percent,  annually  ? 

Without  interest,  the  first  would  have  paid     $236,43. 

the  second,  $21760,44. 

10.  A  merchant  gains  in  trade  such  a  sum,  that  $320  has  the 
same  ratio  to  it,  as  five  times  the  sum  has  to  $2500 ;  what  did  he 
gain  ?  ,qns.  $400. 

.11.  Two  brothers  comparing  their  ages,  find  that  the  sum  of 

18 


206 


(^UESTIOXS. 


both  ages  is  to  that  of  the  elder,  as  19  to  7,  and  to  30,  as  9  to  the 
agf   .f  the  other  ;   what  are  their  ii'^cs  ?  ^       (  21 

^n,.'^    9. 
12.  The  difference  rif  the  sfdes  of  two  square  rooms  j*  to  the  side 
of  the  greater,  as  2  to  6,  and  the  difference  of  th*>ir  square  con- 
tent, is  =  128  feet ;  what  are  the  sides  of  each  of  the^e  rooais  ? 


jins. 


U8. 
(  !4. 

13.  The  profits  of  two  men  !n  their  work,  are  as  8  to  5,  and  the 
proJuct  of  the  numbers  expressins  their  profits  is  360  :  what  was 
the  profit  of  each  ?  ^       {  A"24 

•^"*-  \  Jl5. 

14.  A  merchant  2:aining  $7500  in  6  years,  wjth  a  capital  of 
.$15000,  what  would  he  gain  at  the  same  rate  in  11  years,  with  a 
capital  of  $21000.'' 

15.  It  OIK  man  travels  52  miles  in  a  Jay,  walking  12  hours, 
and  another  61  miles  in  11  hours,  what  will  be  gauged  in 
time,  on  euch,  by  sending  both  at  the  sime  time  to  meet  from 
two  places  180  miles  from  each  other,  to  exchange  dispatches, 
instead  of  ppndin<i  them  each  the  whole  distance  ? 

16.  If  2100  hushels  of  oats  last  200  horse?,  at  a  half  bushel  a 
day,  twenty-one  days,  how  long  will  3700  bushels  last  760  horses, 
at  I  of  a  bushel  per  day  ? 

17.  What  provision  must  be  made  for  an  army  of  9560  men, 
in  bread,  if  they  shall  receive  2  pounds  per  day,  for  70  days,  it 
beiijg  found  by  experience,  that  5000  men  will  need  in  25  days 
SlS.'iOO  lbs.  at  2|  lbs.  per  day? 

18.  If  248  men  in  5  days  of  1 1  hours  each,  dig  a  trench  280  yards 
long,  3  wide,  and  2  deep,  in  how  many  days  of  9  hours  each,  will 
32  men  «lig  a  trenf;h  of  439  yard  long,  6  yards  wide  and  3  deep  ? 
19.  A  has  given  1 15620,  which  laid  in  the  common  stock  6ys.  6  ms. 

jg     ((     (i       "^   921  *'         "         '^         *^         "     4    <'  2  ^^ 

C     "     "       39567,'        "         "        *'         "         "     7    «'  8    « 
D     »'     "       50220,  *'        "         "         '♦         "     8    **  4  '^^ 

E     "     *♦         6943,  "         "        "         "         *'     8   *<  4  " 

The  capital  is  to  be  divided  at  the  end  of  the  time,  and  found  to 
be  double  the  amount  put  in  by  the  stockholJers ;  what  is  the 
share  of  each,  in  the  whole  amount  ? 

20.  Four  merchants  make  ajoint  stock,  under  the  following  ar- 
rangements. A  put  in  $15000,  which  remains  8  years  and  6 
months,  during  which  time  the  association  lasts ;  it  is  agreed,  that 
as  he  is  to  take  the  chief  direction,  he  shall  have  2  per  cent,  pre- 
vious to  all  profits,  besides  his  share  in  the  remaining  profits.  B 
puts  in  $20600  at  the  beginning.  C  puts  in  $l:i200  one  year  after 
the  beginning ;  and  as  he  is  to  work  in  the  partnership,  he  is  to 
have  one  and  a  half  times  the  share  of  the  profits  which  his  capi- 
tal would  entitle  him  to,  if  he  was  not  to  work.  D  joins  two  and 
a  half  years  after  the  beginning,  with  a  capital  of  $60450;  the 
])artnership  being  dissolved,  and  the  whole  profits  made,  being 
|80560,  ^yhat  is  the  share  of  each  ? 


TABLES 

Of  the  Proportional   Subdivisions^   or   Denominate 
Fractions^  of  fVeights^  Measures,  Time,  ^c. 


Explanation. —In  the  rollow.ti*  tables,  the  denominations  of 
the  su*  divisions  will  be  fourui  written  iu  lull,  at  the  beatl  of  each 
table,  and  ni  their  usual  ahhrKvialions  within  the  table? themselves. 
Tb.p  firat  number  ot"  each  square,  's  the  number  ot"  units  of  each 
subiii\  liiion  required  to  mate  the  unit  of  the  kind  touud  at  the  ri§ht 
band;  and  the  lower  numlier  in  ihe  mutae  squnre,  is  the  decimal 
fraction  correspond inij  to  \ho  same  subdivision  and  unit,  carried  to 
7  decimals. 


TIME. 

Seconds. 

Minutes. 

Hourn. 

Day><. 

Years, 

60 

0,016606 

HO 

H. 

I 

3600 
0,0002777 

86400 
0,0011674 

0,<JCn:f444 

24 
0,041666 

D. 

1 

5259487,8 

8766,813 

1>.      H.    M.     S. 

365.  5.48.48 

y. 

1 

CIRCULAR  PARTS. 


Seconds. 

Minutes. 

Degrees. 

Circum- 
ference, 

0,016666 

j 

360. 
0,0002777 

^5,  01 6666          .1 

1296000 
0,00000077 

x>1600 
0, 0000462 

360 
0,002777 

c. 
1 

S08 


S 

U 
w 

o 

h4 


» 

*j  9 

If 

TABIES. 

c^ 

- 

^*  ^ 

§co 

^1 

J 
i 

CO 

CO 

CO 
S  .-^     CO  ^ 

CO 
CO 

o 

CTi 

cT 

1 

CO          -* 

CO             CO 
lO            CO     CO   O 

<>*       _4,    CO           r.00 

O         O    ^  o 
o-^       o 

c2 

p. 

40 
0,025 

320 
0,0031350 

960 
0,0010417 

22144 
0,0004516 

so 

1 

2,75 
0,  3636364 

110 

0, 0990909 

880 
0,0011361 

2640 
0,  0003787 

60n96 
0,0001642 

^ 

''o- 

CO 
lO  CO 

irfco 

o 

220 
0, 0045454 

1760 
0,  0005680 

5280 
0,0001893 

121792 
0,0000821 

o'^ 

CO 

CO 

CO 

CO 

CO  CO 

CO 
o-^ 

16,5 
0,0606061 

660 
0,0015151 

5280 
0,0001893 

15840 
0,0000631 

365376 
0, 0000274 

1 

CO 
«  '-I  o 

o 

o 
o 

00 
00 
CO 

o 

198 
0,0050505 

7920 
0,0001263 

63360 
0,0000158 

1 90080 
0, 0000053 

CO 

^  o 

rt    O 
CO  o 
CO  o 

^ 


TABIJfiS< 


209 


I/} 
«3 


S 

■^:-.  ■{]'■■ 

•i  ^ 

-■••y     i 

J 

a  ^ 

9^ 

•5 

<  -- 

G< 

o  S 
coo 

CO 
CO 

^  o 

c§ 

sS  *^ 

o 

8 

§  CO 

s; 

•*  o 

©<  o 

o 

1 

i.- 

o 

liii 

o 

1 

>.  '-« 

CO 

G^   O 
O  GO 

CO  o 

o 

o-  "  o   '^'  ° 

fi " 

05  ^ 
O 

si 

o 

si 

CO   O 
O  O 

O 

o  ©< 
CO  o 
lO  o 

CO  o 

^  o 
o 

J 

CO 

o 
o 

ii 

O 

CO  O 

CO  O 

^  o 

CO  o 
G^  O 

t-  o 

&<  o 

! 

18* 


nQ 


TABLES* 


CUBIC  MEASURE. 


Inches. 

F..> 

Yards. 

Fathoms. 

IN. 

1728 
0, 0006787 

F. 

1 

46656 
0,00002143 

27 
0, 037037 

Y. 
1 

376648 
0,00000265 

216 
0,  0046296 

8 
0,  125 

FTH. 

1 

CLOTH  MEASURE. 

Inches. 

JVails. 

Quarters. 

Yards. 

Ells. 

IN. 

2,25 
0,  4444 

NL. 
1 

9 
0,  111111 

4 
0,25 

1 

36 

0,  02*^77 

16 
0,  0625 

4 
0,25 

Y. 
1 

45 

0, 02222 

20 
0,05 

5 
0,2 

1,25 

0,8 

E. 
1 

DRY  MEASURE. 


PinU. 

Gallons. 

Pecks. 

Bushels. 

PT. 

8 
0,125 

G. 

1 

16 
0,0625 

2 
0,6 

PK 

1 

64 
0,015625 

8 
0,126 

4 
0,25 

B. 
1 

Eight  Bushels  make  a  Quarter ;  but  as  this  is  not  used  in  any 
part  of  this  country,  any  more  than  the  Wey  and  Last,  we  have 
omilted  them. 


TABLES. 


SIX 


CO 


g 

k; 

^ 

i 

40 

as 

1 

•> 

CO 

CO 

e 

CO 

"% 

s 

^ 

o 

o 

I 

CO 

Q 

CO 
lO  CO 

m 

to 

><i 

oCO 

©<    r 

§> 

T-^    CO 

o 

<o 

15 

o" 

CO 

CO 

uD  CO 

c^CO 
»-i   CO 

CO 

CO 

vt 

CO 

CO 

2J 

ei^ 

o 

CO 
CO  ^ 

CO 

o 

lo 

eo 
o 

l> 

^ 

G^ 

• 

o:) 

rj« 

CO 

1 

o 

O 

CO 

CO 

O  ^ 

^i 

CO  ,^ 

CO  05 
G^   l> 

^  O 

&»  ex. 

kO  CO 
G<   O 

o 

O 

o 

O 

o^ 

o" 

d^ 

o 

o' 

"* 

©^ 

&< 

o 

G^ 

CO 

CO 

Tf 

(N 

eo 

>0 

CO 

I:- 

00 

2g 

r 

a 

. 

CO  Oi 

©<  O) 

CO  a 

-*  Oi 

<y^ 

CO   lO 

^  CO 

CO  G^ 

O  -  1  ^  o 

& 

d^ 

"8 

^8 

CO  o 

o 

^§ 

-s 

f 

»^ 

•^ 

O 

o 

o 

o 

o 

CN 

T— 4 

l_t 

o       o 

CO 

-^ 

CO 

(X            CO 

• 

o 

t^ 

CO 

00 

lis! 

«o 

lO 

G4 

CO  CJi 

^  05 

0^   T}» 

"5 

A.  G^l      »>    CO  i-H     CO  G^ 

o  « 

«>  — 

£ 

O 

GO  O 

lO  o 

CO  o 

o 

-8 

O 

o       o  1 

o 

^  o 

O 

O  1 

o" 

o" 

o 

Habit  alone  determines,  in  different  countries  where  these 
measures  are  used,  to  which  purposes  the  two  different  measures 
of  liquids  are  applied  besides  the  two  liquids  of  which  they  bear 
the  name,  and  these  habits  vary  from  time  to  time.  In  the  state 
of  New- York,  Beer  measure  is  little  u^ed,  but  the  ordinary  mea- 
sure for  all  liquids  is  Wine  measure. 


u^ 


TABLES. 


BEER  MEASURE. 


Pints 

Quarts     Ua'iotis. 

■  .arr  ,s 

Ho^^sheinh 

Buiti, 

2  P. 
0,5 

1 

8 
0,125 

4                    G 

0, 25     .       1 

288 
0, 00347 

144      j       36 
0,00o    4  0,02777 

B. 

1 

4<>2 

n  H       '        r)4 

1,5 

1    HHD. 

864 

432      1      108 

3 

2 

iB. 

TROY  WEIGHT. 

(Used  for  gold,  -liver,  jewels,  and  retail  dealing.) 


Grains.     Pennyweights,  \  Ounces. 

Pounds. 

GR. 

24 
0,0416660 

DWT. 
1 

480 
0, 0020833 

20 
0,05 

1 

5760 
0,0001736 

240 
0,0041666 

12 
0,08333 

LB. 

1 

APOTHECARIES  WEIGHT, 


(Used  in  conij.iiuudit.^  ri, 

.-diciues.) 

Grains 

Scruples. 

Drums 

Ounces. 

Pounds. 

GR. 

20 
0,05 

sc. 
1 

60 

0,016666 

3 
0,  333 

DR. 

1 

480 
0,  0020833 

24 
0,041666 

8 
0,  125 

OZ. 

1 

5760 
0,0001736 

288 
0, 0034722 

96 
0,0104165 

12 
0,08333 

LB. 

1 

TABLES. 


Sl<5 


AVOIRDUPOIS  WEIGHT 

. 

Drams 

Ounces. 

Pounds 

(^uortrs.  \iwt. 

Tons. 

DR. 

16 
0, 0626 

oz. 

1 

266 
0,0039014 

18 
0, 06566 

LB. 

1 

7168 
0,0001396 

448 
0,0022321 

28 
0,0367143 

an. 

1 

28672 

179^' 
« »,  0005680 

112 

0,00P0«78 

4 
0,26 

nwT. 
1 

673440 

35840 

2240 
0, 0004464 

80 
0,0126 

20 
0,06 

TONS. 

1 

This  kind  of  weight  is  used  in  every  other  case  of  mercanfile 
transaction,  whether  in  the  great  transactions  of  general  com' 
merce,  or  in  the  retail  trade. 

oz.  dwl.  gr. 
1  lb.  Avoirdupois  =    14.  11.  16     Troy. 
1  oz.  "  =  18.    61      ♦« 

1  dr.  ♦*  =  1.    3i      '* 

Before  the  last  law  in  England,  of  1826,  regulating 
weights  and  measures,  the  following  were  the  cubic 
COHtents  of  the  dilBferent  measures  of  capacity  ;  viz  : 

The  Bushel,  2160|  cubic  inches  =  a  cylinder  8  in. 
deep,  18,6  in.  diameter. 

The  Gallon,  dry  measure,  268|  cubic  inches. 
»*         "         for  beer,  282       "  " 

<*  "         for  wine,  231        »'  <* 

These  two  latter  gallons  have  to  each  other  the  same 
ratio  as  the  weights  of  Avoirdupois  and  Troy. 
By  the  law  of  1826, 

The  Bushel  contains     2217,6  in.  cubic. 
The  Gallon      ''  277,2        ** 

and  is  used  indiscriminately  for  dry  and  liquid  measure. 
The  capacities  are  determined,  not  by  measurement 
of  the  cubic  contents,  but  by  the  weight  of  pure  water 
at  the  temperature  of  62°  of  Fahrnheit's  thermometer 
contained  in  the  vessels  ;  the  bushel  holding  80,  and  the 
gallon  10  lbs.  avoirdupois. 


S14 


OTABLES. 


a  •- 

<n 

"■    O 


r 


"5 
a  2 

*.  a 

O    3    « 

la  Is  « 
^  >  ^ 


SJ  «  ° 

M  a  t- 
SB  *   ^ 

a>  5 

*J  ♦J 
iS  6/5 
C/2  ,S 


(^ 
1 

e 

1 

s 

c 

c 

'J 

■1 

1 

f            Si          f  1 

cr! 

1 

'^       g£       i             £    £    :2 
!  —       r-.  ?      i             ^    ''^    S 

1 

^ 

1  £       V  ^-      :              .      .    ^ 

•  o      -■  -       :            £    •£    =o 

;l  5^  ;.     Sis 

3 

^    ij^        :  :       :      1          i    i    : 

^^r-^?|    i  i.  — :  !        j   ;   • 

c2      a.  -  j:  -3  c»,  :.-    ■  i:^  ai        5?    f-  ci  a?  0  O  ail 

■8,- 

si 

|2 

5^  rj      J^ 

I' 

3 

S 

Value  of  Foreign  Coins  accoHmg  to  the   Laws  of  the   United 

Sfiites. 

Gold  coins  of  G.  Britaio  aud  Portugal  are  rated  at  ;JJ  1  for  27  grainj- 
"  France  »'  "  27|     '> 

»'  SpaiQ  "  "  28J    " 


TABLES. 


215 


Coiirtc  of 

All      't  Jam, 
Antwerp, 

ditto, 
Au^^burg-, 

ditto, 
Basil  and  i 
Zuric,        \ 
Berliu, 
ditto, 
Boloj^na, 
Coustan-    ) 
tinople,      \ 

ditto, 
Copenhageu 

ditto, 
Frankfort  ) 
OD  Main,   \ 
ditto, 

Genoa,      < 

Geneva, 
Hamburg  t 
£l  Altona,  S 
Lci!!zi;^&  ) 
Dresden,  \ 
Lfsbou, 
Leghorn, 

London, 
Milan, 

Naples       ? 

New-York, 

Palermo, 

Petersburg, 

Spain, 

Stockholm, 

Turin, 

Venice, 

Vienna, 


1 

*? 

gtMS 

§ 

or 

c«. 

rtc'^s 

^"^ 

r 

3 

r 

3 

s: 

10<3 

r 

3U0 

^ 

" 

rorg 

luO 

V 

JuO 

^ 

jt;  i 

r 

3 

r 

300 

cr 

1 

r 

300 

1 

^1 

30. 
100 

g 

100 

s 

I 

100 

100 
3 

V 

iOO 

r 

3 

1 

1: 

I 
3 

^•■ 

,  3 

] 

(Q 

L 

i 

iC 

1 

r 

1 

r 

1 

1 

& 

(  1 

r 

•  > 

r 

3 

r 

-JUO 

\l 

1 

1 

2-_      Denomtnatioti. 


Fr»ar«,  .  . 
Francs,.  ., 
Fiaiics,  .  . 
P.anrN.  . 
Florin  Ct, 


Francs, 

Frrmcs, 

B;-«nco  Prii&siau, 
Fraucs, 


Frniics, 

Piastre,  

Francs, 

Rix  dollar,. 

Francs, 

Fraiirs, 

Francs, 

Piaster  (ot  .15s.  h 

.b.) 

Livres  courant,  .. 

vlark  banco, 

Fr-\iics, , 

Francs,. , 


Francs, 

P)  istre(of8Reals) 

.ij  sterling, 

Francs, 

Francs,      

Franc?, 

Lire  iinue«-iale, .. 
Ducat  (of  10  Car- 

lin.,) 
Franc, 
Friioc,. 
R^ible, 


Pistole(of  32  reals) 
Piastre  (of  8  reals) 

Francs, 

Francs, 

Francs,. 

Frnno, 

Florin,  


lily. 


.70 

5.->, ! 
99,50 
U7 

25. 

^9,50 

78 

56 

102,5 

2,90 

♦35,5 

4,45 
79 
99,5 
99,5 

4,8< 

166 
190 
24 

!  76 

480 

5,10 
23 
21 
55 
71irelo 
l.OS 


4,20 

0,1 8i' 

46 

4,40 

15 

3,  75 
25 
50 
61 
23 

2,53 


Denomination. 


Deniers  groats. 
Ueniers  groats. 
Francs. 

Florins  courant. 
Centimes. 

Francs. 

R-ix  dollars. 
80  Francs. 
Sols. 

Piastres. 

Francs, 

Centimes. 

Francs. 

Rix  dollars. 

Francs. 

Francs. 

Francs. 

Francs. 
Francs. 
Sols  Lubs. 

Rix  dollars. 

Rees. 
Francs. 
Francs. 
Pence  sterling. 
Sols  imperial. 
Sols  Courant. 
Francs. 

Franc?. 

f^ollars. 

Grains. 

Francs, 

Francs. 

Francs. 

Shillings. 

Sols  Piedmont. 

Ducats  banco. 

Cruezers. 

Francs. 


END. 


TABLE  OF  CONTENTS. 


Introduction.  Pcig^' 

Part  I. — First    Elements  and  Deductions  of  the 

Four  Rules  of  Arithmetic. 
Chapter  I. — Fundamental  Idea  of  Quantity.     Sys- 
tem of  Numeration,      ...  0 
I     "     II. — General    Ideas  and    Notation  of  the 

Four  Rules  of  Arithmetic,    .         .         13 
"     III. — Four  Rules  of  Arithmetic  in  Whole 

Numbers,    .....  18 

"     IV.— Of  Vulgar  Fractions,      ...         39 

"     v.— Of  Decimal  Fractions,      ...         63 

"     VI.— Of  Denominate  Fractions,        .         .         76 

Fart  II. — Practical  Applications  of  the  Four  Rules 

of  Arithmetic. 
Chapter  I — General  Principles  of  the  Application  of 

the  Four  Rules  of  Arithmetic,        .         91 
"       II.— Application  of  the  Four  Rules  of  Arith- 
metic to  all  kinds  of  Questions  involv- 
ing Fractions  of  either  kind,  .  98 
Fart  III. — Of  Ratios  and  Proportions. 
Chapter  I. — Elementary  Considerations  of  Ratio,       108 
"       II. — Arithmetical  Proportion,        .         .          113 
"      III. — Geometrical  Proportion,          .          .          116 
"      IV —Rule  of  Three,      ....         124 
"       v.— Compound  Rule  of  Three,     .         .         136 
"      VI. — General  Application  of  Geometric  Pro- 
portion,       144 

Part  IV. — Extension  of  Arithmetic  to  higher  Branches 

and  other  practical  Applications. 
C/jflp<«r  I.— Of  Square  and  Cube  Roots,     .        .  150 

"  II. — Of  Progressions  or  Series,  .  .  165 
"  III. — Of  Compound  Interest.  Ideaof  Annuity,  180 
"     IV.— Of  Alligation,  or  Mixture  of  Objects  of 

different  Value,  .  .  .  187 
Collection  of  Questions,  .  .  .  ,  193 
Tables,  .         , 207 


i 


14  DAY  USE 

RETUKM  TO  DESK  FROM  WHICH  BORROWED 

LOAN  DEPT. 

This  book  is  due  on  the  last  date  stamped  below,  or 

on  the  date  to  which  renewed. 

Renewed  books  are  subject  to  immediate  recall. 


CLF  m ! 


JAN  2  0  1966  6  > 


m&'o  Lii 


m^Mm^iAm 


LD  21A-60m-3.'65 
(F2336sl0)476B 


Genei. 
University  o^ 

Berkele. 


-  / 


YB  17369 


^ 

9181  i^l 

^}il6Z. 

^57 

THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 

